Growth, Trade and Economic Institutions

Growth, Trade, and Economic Institutions Edited by Tapio Palokangas Bjarne S. Jensen Dieter Bös † Giacomo Corneo Reino ...

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Growth, Trade, and Economic Institutions Edited by Tapio Palokangas Bjarne S. Jensen Dieter Bös † Giacomo Corneo Reino Hjerppe Juha Honkatukia Journal of Economics Supplement 10

SpringerWienNewYork

Univ.-Prof. Dr. Tapio Palokangas Department of Economics, University of Helsinki, Finland

Univ.-Prof. Dr. Bjarne S. Jensen Department of Economics, Copenhagen Business School, Denmark

Univ.-Prof. Dr. Dr. Dieter Bös † Univ.-Prof. Dr. Dr. Giacomo Corneo Department of Economics, Free University of Berlin, Germany

General Director Dr. Reino Hjerppe Dr. Juha Honkatukia VATT, Helsinki, Finland This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. © 2005 Springer-Verlag/Wien Printed in Austria SpringerWienNewYork is part of Springer Science + Business Media springeronline.com Typesetting: Camera-ready by authors Printing: G. Grasl Ges.m.b.H., 2540 Bad Vöslau, Austria Printed on acid-free and chlorine-free bleached paper SPIN: 10920107 With 11 Figures Library of Congress Control Number: 2004117221

ISSN 0084-537X

ISBN 3-211-00793-8 SpringerWienNewYork

Preface The seven papers in this supplement issue were originally presented at the eighth conference of Dynamics, Economic Growth, and International Trade (DEGIT) in Helsinki, Finland, May 30-31, 2003. The DEGIT VIII conference was organized jointly by Copenhagen Business School (CBS), Department of Economics University of Helsinki, Department of Economics Government Institute for Economic Research (VATT) Financial support from VATT and the Central Bank of Denmark is gratefully acknowledged. This conference is a sequel to seven previous conferences with the same theme held in Denmark (1996), Hong Kong (1997), Taiwan (1998), Tilburg (1999), Rome (2000), Vienna (2001), and Cologne (2002). This series of conferences is a forum for economists working in the ﬁeld of growth and trade theory. For previous DEGIT publications, see Review of International Economics 7(3), 1999, Review of Development Economics 5(2), 2001, and German Economic Review 4(1), 2003. It is our sad duty to report that the sixth member of our Editorial Board, Dieter B¨os, the Managing Editor of the Journal of Economics, died in Bonn, June 20, 2004.

January 2005

Tapio Palokangas and Bjarne S. Jensen (Special editors)

Contents Jones, R.W., Kierzkowski, H.: International Trade and Agglomeration: An Alternative Framework . . . . . . . . . . . . . . . .

1

Jensen, B.S., Larsen M.E.: General Equilibrium Dynamics of Multi-Sector Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Palokangas, T.: Common Markets, Economic Growth, and Creative Destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Grieben, W.-H.: Schumpeterian Growth and the Political Economy of Employment Protection . . . . . . . . . . . . . . . . . . . . . . .

77

Grossmann, V.: White-Collar Employment, Inequality, and Technological Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119

Goenka, A., Poulsen, O.: Indeterminacy and Labor Augmenting Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

Olsson, O.: Geography and Institutions: Plausible and Implausible Linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

J. Econ. (2005) Suppl. 10: 1-16

International Trade and Agglomeration: An Alternative Framework Ronald W. Jones and Henryk Kierzkowski Received September 1, 2003; Revised version received August 16, 2004 c Springer-Verlag 2005

The New Economic Geography, as expounded by Krugman and others, highlights the importance of greater extent of agglomeration as incomes grow in the world economy. In this paper we suggest an alternative framework in which, as incomes grow, greater degrees of economic fragmentation and dis-agglomeration are encouraged. In both approaches increasing returns to scale are crucial, but in ours these are found in service sectors that allow separate production blocks to be coordinated instead of being found within separate production blocks. Keywords: agglomeration, fragmentation, service links, production blocks JEL classiﬁcation: D20, F12, R30

1 Introduction In the vast array of ﬁelds in economics, international trade and economic geography should be neighbors sharing similar interests and preoccupied with a strongly overlapping range of issues. Alas, one could say that the scientiﬁc telescopes of each specialization had been trained for a long time in diﬀerent directions. This state of isolation could not last and either an international trade economist would discover that commerce, within or across countries, involves geography; or a geographer would have observed that trade is one of the best examples of spatial displacement. In the event, Paul Krugman was the ﬁrst to seize the connection in a 1991 Journal of Political Economy paper and has been running with the main idea ever since.1 Other trade economists soon saw a new opening 1 In the economics literature concern over spatial aspects was expressed earlier by Stephen Enke (1951) and Paul Samuelson (1952). Samuelson remarked,

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and a way of enriching their discipline. Having “discovered” geography, international trade economists had no hesitation telling economic geographers how their ﬁeld really should be structured and developed. As in the case of the “new” trade theory, the breakthrough in the “new” economic geography has come from the application of increasing returns to scale, especially in the context of monopolistic competition utilizing the functional form made famous by Avinash Dixit and Joseph Stiglitz (1977). Increasing returns to scale could not alone do the trick of re-orienting the ﬁeld of economic geography; in addition to increased realism, transportation costs have been called in to give the new models an increased complexity in order to generate interesting results. There is no doubt that the tools and the analytical machinery developed in the course of the new trade revolution have proved very popular. However, it seems pertinent to ask what purpose is served by applying them to geography. A convincing answer would be that the traditional economic geography framework and the tools it employed were not capable of explaining the existence of some important stylized facts. Recall that the existence of intra-industry trade and the alleged inability of the traditional international economics models to explain this trade, ushered in the “new” trade revolution almost two decades ago. Was there, more recently, a corresponding stylized fact related to economic geography that escaped explanation until the increasing-returnsto-scale artillery was brought in? The question of the agglomeration of economic activities or its opposite could be thought of as requiring proper modeling and explanation. It is not obvious, however, that there is some overwhelming empirical evidence demonstrating clear trends regarding agglomeration (or dis-agglomeration) on a country, regional or global basis. The desire to understand the mechanisms driving these “Spatial problems have been so neglected in economic theory that the ﬁeld is of interest for its own sake.” (p. 284). In 1976 a symposium was held in Stockholm, and the Foreward to the publication of the Proceedings (see Bertil Ohlin, Per-Ove Hesselborn and Per Magnus Wijkman, eds, 1977) commented, “A special aim of the symposium was to bring about an exchange of ideas between economists and economic geographers interested in trade and movements of the factors of production.” The “new” economic geography has expanded in leaps and bounds. Even reviews of the existing work have become numerous. Among most comprehensive surveys are: Neary (2001), Martin and Sunley (1996), Sunley (2001), Sheppard (2001) and Urban (2001).

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processes may provide suﬃcient justiﬁcation for an interest in this subject.2 There exists evidence showing that the global economy does not consist of a single core or even a limited number of centers and peripheries. Instead, the world economy becomes an increasingly, even though not evenly distributed, complex industrial structure spanning not only individual continents but the entire globe. International production networks have emerged in a manifold of industries and products: sports footwear, mobile phones, cars, clothing, computers, and furniture to name only a few. While there obviously are agglomeration forces operating in some areas, dispersion of economic activities is also a fact of life. One of the consequences of dis-agglomeration manifests itself in a rapid expansion of international trade in parts and components. The recent empirical study by Francis Ng and Alexander Yeats (200l) shows this new phenomenon for East Asia. Between 1984 and 1996 East Asian imports and exports of manufactured components grew annually between 2 and 3 times as fast as imports and exports of traditional production. It is highly probable that the trade in parts and components also trumped intra-industry trade. The maquiladora phenomenon also shows that dis-agglomeration of production takes place in the U.S. - Mexico context.3 Further north, Canada and the United States had undertaken sharing of production many years ago, especially in the automobile industry. More recently, India has emerged as a powerful attractor for a range of intermediate activities in manufacturing and services. Europe is certainly moving in the same direction.4 All in all, it has been estimated by Yeats (2001) that recently about 30 percent of global manufactured goods trade takes the form of trade in parts and components. Corresponding numbers for the 1950s and 60s do not exist, but they surely must have been very small indeed. It follows that growth of intra-industry trade must have been outpaced by a new type of trade associated with dis-agglomeration. 2 One of the critics of the new economic geography models based on the Dixit-Stiglitz approach takes Masahisa Fujita, Paul Krugman, and Anthony Venables, the authors of The Spatial Economy (1999), to task:

“While such formal modeling may increase the credibility and popularity of the ideas with the economists, it provides no evidence as to their actual empirical signiﬁcance and their initial assumptions appear mainly to be made on the basis of modeling convenience rather than with any regard to empirical relevance.” [Sunley (2001), p. 136.] 3 Interesting work on the phenomenon of outsourcing in the Mexican context is found in Robert Feenstra and Gordon Hanson, 1996. 4 Interestingly enough, the grand project of a supersonic passenger plane, Concord, conceived primarily by the French and British back in the 1960s could have been one of early examples of dispersion of production across Europe.

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The phenomenon of international production networks and trade in parts and components reinforces the importance of transportation costs stressed in the new economic geography. But what kind of transportation costs - producers-to-consumers or producers-to-producers? Surely it must be the latter in a world where production of a pair of jeans can be broken down into 24 stages and allocated among Pakistan, Mainland China, Hong Kong and Malaysia with more than a dozen border crossings being executed before the ﬁnal product is shipped oﬀ to consumers.5 Are producers-to-consumers transportation costs so important as to neglect producers-to-producers transportation costs? It would seem likely that the industrial landscape generated by our theoretical models looks diﬀerent depending upon which transport links are brought to the fore of the analysis. As Peter Neary (2001) has recently commented in discussing the appearance of The Spatial Economy (1999) by Masahisa Fujita, Paul Krugman, and Anthony Venables, “New economic geography has come of age.” While recognizing its important and numerous contributions, we should advance other explanations of phenomena arising in the common grounds shared by two neighbors - economic geography and international trade. 2 Two Alternative Scenarios In suggesting an alternative framework for examining whether economic growth is accompanied by a greater degree of agglomeration or, instead, by a spread of productive activity or dis-agglomeration, we alter somewhat the focus provided in the recent work of Krugman and others 5 The Economist of December 27th 2002 reports on a rapid growth of logistic companies operating on a global scale.

“So what exactly can smart logistics do for companies? One example is TPG’s contract with Ford to service its Toronto factory. This plant produces 1,500 Windstar minivans a day. To keep it running virtually round the clock, TPG has to organize 800 deliveries a day from 300 diﬀerent part makers. Its software must be tied into Ford’s computerised production system. Loads have to arrive at 12 diﬀerent points along the assembly lines without ever being more than 10 minutes late.” But TPG is a traditional freight company. The new type of logistics company, such as Exel, does much more. It is like the chef of an orchestra. The Economist continues: “One of Exel’s biggest contracts is with Ford, for which it organizes supplies for seven factories around Europe. Exel also works for Volkswagen in its operations in Spain and Mexico. The factory in Puebla, Mexico, turns out 1,400 new Beetles a day. And Exel helps with Nokia’s logistics as well, especially in China and South-East Asia.”

International Trade and Agglomeration: An Alternative Framework

5

in order to provide a benchmark comparison. We leave out of account the question of costs involved in having produced ﬁnal goods reach the consumer and, instead, enquire about the possibility of breaking an integrated production process into separate fragments that could be located in other areas or other countries. In asking about links among producers instead of between ﬁnal producers and consumers we also dispense with the need for utilizing the Dixit-Stiglitz utility function to express taste patterns for consumers facing an array of diﬀerentiated ﬁnal goods. This function has provided yeoman service in “new trade theory”, growth, and political economy in supplying a foundation at the micro level for Chamberlinian monopolistic competition with ﬁrms operating in the range in which increasing returns to scale are still found. Increasing returns are essential in our alternative scenario, but they are assumed to take place in what we call service link activities such as communication and co-ordination services that are required to establish a functioning network among fragments of production blocks that are located in diﬀerent geographical locations (Ronald Jones and Henryk Kierzkowski, 1990, 2001a). These service activities include (but are not limited to) transportation services, the ones that play such a crucial role in the “new economic geography”, albeit assumed there to be of the constant-cost variety. Herein lies a crucial distinction with our alternative: Increasing returns are assumed to reside in service link activities (including transportation) instead of on the factory ﬂoor (within production blocks). In section 4 we suggest what empirical work has to say about the nature of service links. In each of the two alternative scenarios we now present we compare the costs of producing a ﬁnal commodity when an integrated production location (or ﬁrm, IF) is used as opposed to having the production process split into two fragments located in diﬀerent regions or countries, perhaps produced by two diﬀerent ﬁrms. If such a split occurs, costs of production (neglecting transport costs or other coordinating service link activities) are lowered since it is possible to select locations such that factor prices and/or factor productivities are for each fragment more suited to factor proportions in that fragment. Regions in which labor is relatively inexpensive are used for the more labor-intensive fragment. For example, Nike, in making sports apparel, does the design work in the United States but outsources almost all the actual production activity to ﬁrms in Asia. Likewise, the Swedish furniture ﬁrm, Ikea, early on sent its actual production activity to Poland and used its Swedish labor force to design the individual pieces. In both Fig. 1 and Fig. 2 (pages 6 and 7) ﬁnal output, Y, is shown on the horizontal axis and a pair of total production cost loci are drawn, labeled IF when all activity takes place in a single location with one ﬁrm, and OF (outsourced fragments) when the costs in the two separate fragments

6

Ronald W. Jones and Henryk Kierzkowski Costs

O

C

TF

IF

B OF

IT

A 0

Y0

Y (output)

Fig. 1. Agglomeration. IF: Integrated Firm. OF: Outsourced Fragments (Production). IT: “Iceberg” Transportation. TFO: Total Fragmented Operation. ABC: Minimum Cost Schedule.

are added up (production costs only). These two fragments provide the appropriate balance of necessary output for any value of Y. If production is split between two fragments located in diﬀerent areas, these fragments must be brought together and coordinated, thus incurring extra costs of transportation, communication, and obtaining knowledge of where best to locate the fragments. These service link costs tend to be higher if fragments are located in diﬀerent countries than if they are merely placed in diﬀerent regions of a single country. Where Fig. 1 and Fig. 2 diﬀer is in the kind of activity in which increasing returns are found. Fig. 1 characterizes our version of the assumptions made in the Fujita, Krugman and Venables model, in which increasing returns are found within production blocks. The simple way of modeling such increasing returns is to combine ﬁxed costs (along the vertical axis) with constant marginal costs (shown by the slope of the total cost curve), leading to the rising IF and OF loci in Fig. 1. (Note that with two regions from which to chose, the costs of production along OF are everywhere lower than along the IF-locus if production is positive). Following their treatment we assume that the entire costs of linking the two fragments together is in the form of transport costs between fragments where the so-called iceberg model of such costs is used. (The iceberg analogy was introduced by Paul Samuelson in his 1954 Economic Journal discussion of the international transfer problem

International Trade and Agglomeration: An Alternative Framework

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Costs C

IF

B

OF

O TF

SL 0

Y0

Y (output)

Fig. 2. Dis-Agglomeration. IF: Integrated Firm. OF: Outsourced Fragments (Production). SL: Service Link Costs. TFO: Total Fragmented Operation. OBC: Minimum Cost Schedule.

when transport costs exist.6 ) In that scenario, a unit of output exported from one locale will arrive at a diﬀerent locale diminished in size, much as part of an iceberg would melt if transported from one region to another. The crucial aspect to notice is that this makes transportation a constant-returns-to-scale activity - doubling the output transferred between locations will double the loss eaten up in transport. There is no doubt that such an assumption is useful in avoiding separate activities whereby factors of production are combined to produce the services of transportation. However, it introduces a form of service link in Fig. 1 not matched in Fig. 2’s alternative. The TFO (total fragmented operation) costs are found by adding the OF locus to the ray from the origin representing transportation costs. In the alternative portrayed in Fig. 2, constant returns to scale are assumed both for the integrated production block (IF) and for the costs (production only) of the combined activities for the separate production blocks (OF), which are lower because each fragment is located in an area in which there is a better match among factor prices, technology, and factor proportions. Service link costs are required in order to co-ordinate the outputs of the separate fragments, and we make the 6 As a matter of fact, Michael Rauscher has pointed out to us that the grand master of economic geography, Johann Heinrich von Thunen, 130 years previously, described how a horse that transports wheat from the country side to a city market eats a fraction of it en route.

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extreme assumption that all such costs are constant regardless of the scale of activity. Thus the TFO cost schedule is a shifted-up version of the OF locus. In each diagram the cost of the best mode of production is shown by a broken heavy line, with the break appearing at output level Y0 . Do larger scales of output encourage or discourage agglomeration? The contrast between the two scenarios is striking. In Fig. 1, the characterized version of the Fujita, Krugman and Venables model with iceberg transportation costs, disaggregated output in two locales is appropriate for small levels of output, up to Y0 . Up to this point the costs of connecting the two fragments by incurring transport costs are outweighed by the beneﬁts of lower marginal costs in each fragment, but for higher levels of output it pays to combine all output in an integrated production block subject to increasing returns to scale and thus to obviate the need to pay for transportation. By contrast, in Fig. 2 it is a large scale of output (greater than Y0 ) that encourages dis-agglomeration. With increasing returns found in the service link activities (including the costs of transportation), it pays to outsource the originally verticallyintegrated production process into two fragments in diﬀerent locales especially well-suited to their factor proportions. In Section 4 we shall dwell more extensively on the nature of the costs of service links, including transportation. There is general agreement that there have been signiﬁcant technological improvements that have brought about a lowering in the costs of service links. This is especially true of the costs of communication, which now have almost reached the vanishing point. It also holds, to a lesser extent, with the general costs of obtaining information and, as well, costs of transportation between producing regions. The consequences of such downward shifts in these costs in the two alternative settings are profound. In Fig. 1, a lowering of the iceberg transportation costs causes the (TFO) schedule of costs of total fragmented operation to rotate in a clockwise direction from the initial point on the vertical axis. As a result, the change-over point of output, Y0 , moves to the right. In Fig. 2, a technologically-inspired reduction in the costs of service links shifts the TFO schedule downwards, causing the change-over level of output, Y0 , to move to the left. The result: In both cases the range of outputs in which dis-agglomeration is the preferred mode of production increases. In Fig. 1 as output expands the consolidation of production in one location is delayed, whereas in Fig. 2 the desirability of fragmented production occurs at an earlier stage of growth. One of the widely-recognized asymmetries on the international scene is the greater ease of moving commodities and middle products (including physical capital) among countries than it is to move labor. The large international migrations of labor witnessed at the end of the 19th cen-

International Trade and Agglomeration: An Alternative Framework

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tury and beginning of the 20th century seem now a thing of the past, although some migration, both legal and illegal, still takes place. This asymmetry provides the basis for the doctrine of comparative advantage and suggests limits on the degree of international agglomeration that can be expected. Whereas within a country relatively mobile labor can aid and abet a process of agglomeration into a few urban nodes, on a global scale we have witnessed a greater degree of outsourcing of fragments of the production process in a manner reﬂecting both the Ricardian and Heckscher-Ohlin rationale for the nature of trade. Such dis-agglomeration among countries is encouraged by the increasing returns to scale found in many service link activities as world incomes rise, by the signiﬁcant improvements in technology in the service area, as well as by a general lowering of regulatory barriers to international trade. With labor relatively immobile between countries, the doctrine of comparative advantage guarantees that no country will empty out as a consequence of the forces of agglomeration.

3 Fragmentation May Encourage Agglomeration and Growth In the preceding section we have suggested that greater levels of output in an industry tend to encourage a fragmentation of a verticallyintegrated production process, with outsourcing reaching even beyond a nation’s borders. Suppose such a process is taking place not only in a single sector but also in many industries world-wide. Then it is possible to argue that fragmentation may provide a stimulus to subsequent agglomeration at a global level! The argument for such a possibility rests in part on what we have termed the “horizontal aspects of vertical fragmentation” (see Jones and Kierzkowski, 2001b). Suppose that in a number of industrial sectors economic growth, technical progress, increasing returns in connecting service-link activities, and deregulation eﬀorts have all conspired to promote a fragmentation, both locally and internationally, of production processes. We assume that some of these fragments more closely resemble each other in an inter-industry comparison than do the original integrated activities. This encourages further technical progress serving to make such fragments even more uniform and useful in a number of diﬀerent sectors of the economy. (Consider the spread in the use of computer chips from computers to a wide range of uses ranging from toasters to automobiles). Furthermore, the overall techniques of production (or factor proportions) of such fragments may be rather similar. All this serves to encourage an agglomeration of a new industry producing such fragments for a wide array of sectors both locally and internationally.

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The time-honored arguments about labor with certain skills being attracted to a center where a variety of productive activities require such skills seem appropriate in this setting. Alfred Marshall (1890) was an early exponent of the kind of externalities that may emerge when fragments of diﬀerent industries share somewhat similar factor proportions and types of labor skills are located in the same region. In his words, “The mysteries of trade become no mystery; but are as it were in the air. Good work is rightly appreciated, invention and improvements in machinery, in processes and general organization of businesses have their merits promptly discussed: if one man starts a new idea, it is taken up by others and combined with suggestions of their own; and thus it becomes the source of further new ideas.” This Marshallian view has been picked up by other geographers. Thus, “Economists since Alfred Marshall have argued that cities facilitate the ﬂow of ideas. However, most urban research focuses on the role that the density plays in reducing transportation costs between suppliers and customers (Krugman, 1991). However, Dumais et al. (1997) show that manufacturing ﬁrms in the USA since 1970 have not based their location decisions on the presence of suppliers and customers. Instead, ﬁrms locate near other ﬁrms that use the same type of workers.”7 These ideas are consistent with a Heckscher-Ohlin basis for trade between countries based on diﬀerences in factor endowments and factor requirements in production, even without strong elements of imperfect competition. It would not be diﬃcult formally to model externalities whereby factor productivities in one sector are positively aﬀected by increases in activity of similar factors in other sectors closely related in their input and skill requirements. The process of agglomeration outlined above presents an alternative to the agglomeration story told by the new economic geography. Perhaps it follows more closely what traditional economic geographers 7

Edward Gleaser (2000), p. 84. Others expressed similar views. Martin and Sunley (1996) state on pages 285-6, “one of the most important limitations of Krugman’s geographical economics is his stubborn concentration only on those externalities that can be mathematically modelled, and thus his reluctance to discuss the geographical impacts of technological and knowledge spillovers. The recent geographical literature has begun to assign key importance to technical change and technological externalities in shaping and transforming the space economy”

International Trade and Agglomeration: An Alternative Framework

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tell us is important. It presents a more complex world in which fragmentation of production results initially in a dispersion of economic activities. However, as a realignment of production patterns takes place within and across countries, the forces of agglomeration once again are in evidence. The addition of these externalities allows the forces of fragmentation and subsequent agglomeration to become engines of growth.8 As fragmentation and technical progress take place, the locales in which such pressures for agglomeration occur may change over time. An example of this was provided years ago in Ray Vernon’s discussion of the product cycle (1966). At early stages of production a search is under way for the best techniques of producing a new product. In the face of uncertainty, the location of production is guided largely by the existence of labor (and perhaps capital) of a number of diﬀerent skills. This leads (so Vernon argued) to a locale in an advanced country such as the United States. Eventually things settle down, and a technology requiring relatively heavy use of less skilled labor is formulated. As a consequence, the industry moves to a less developed country in which such labor is relatively inexpensive. There is another route whereby fragmentation may eventually lead to a greater degree of agglomeration. The international fragmentation of production blocks is only made possible by the use of connecting service links, and it is in these service activities that we have argued that strong increasing returns are found. Just as in the arguments put forth by Krugman and others that increasing returns foster agglomeration, such agglomeration might be expected within the service sector. Indeed, this seems to be the case in activities such as ﬁnancial services and insurance. Deregulation should certainly be given much of the credit for allowing more open trade in services, and one of the by-products of international fragmentation and its role in dis-agglomeration is the tendency for activities that provide service links themselves to agglomerate. This is consistent with the observed concentration of many service activities in advanced countries and international outsourcing of production blocks to less developed areas.

4 Increasing Returns and Technology in Service Link Activity Where are increasing returns to be found? The crude assumptions that we have made is that they are found exclusively in the service link activities that facilitate a coordination of fragmented production blocks as opposed to constant returns to scale within such blocks. We need 8 For somewhat similar views see the discussion of “clusters” as providing externalities and inﬂuences on productivity growth in Michael Porter (2000).

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not rely on such a stringent dichotomy. Instead, we would argue that the kind of economic activities that are most often associated with increasing returns are ones in which economic information is gathered, where ﬁnancial aids to trade are obtained, where shipments are insured, where communication between locations far apart are required, and even where transportation activities are involved.9 Although some service activities are found within production blocks, it seems diﬃcult to ﬁnd strong evidence of increasing returns in actual production. In a relatively recent estimate of production functions in the United States, Susanto Basu and John Fernald (1997) have come to the following conclusion: “A typical (roughly) two-digit industry in the United States appears to have constant or slightly decreasing returns to scale” (p. 249) and furthermore “most plants and engineering studies ﬁnd essentially constant returns to scale.” (p. 263). The evidence from Canadian manufacturing industries provides some support for the opposite view that there are increasing returns to be found in production.10 Technological progress in service link activities, including transportation, has been an impressive feature of the past few decades. For example, one can hardly discuss global production without making reference to telecommunications. It is generally accepted that launching telecommunications services involves high ﬁxed costs. By contrast, the marginal cost is miniscule. In a bygone era telephone operators struggled to make speedy connections. Today a user dials a number practically anywhere in the world and is put through immediately. Long-distance transmission of text or images has also become much less costly. Sending production plans from England to Singapore became signiﬁcantly easier two decades ago with the introduction of Fax technology and DHL. Today it is possible to transmit text and images in living color instantaneously and almost costlessly. 9

Alfred Marshall (1890) has this to say about transportation: “A ship’s carrying power varies as the cube of her dimensions, while the resistance oﬀered by the water increases only a little faster than the square of her dimensions; so that a large ship requires less coal in proportion to its tonnage than a small one. It also requires less labour, especially that of navigation. In short, the small ship has no chance of competing with the large ship between ports which large ships can easily enter, and between which the traﬃc is suﬃcient to enable them to ﬁll up quickly.”

10

Michael Benarroch (1997), p. 1084.

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5 Concluding Remarks Both “old” and “new” geographers have cited many reasons why economic activity is not spread uniformly within a country or, indeed, among countries. Despite the standard economic doctrine of diminishing returns, many reasons can be cited for a “bunching up” of productive activities and residences. Individuals desire to consume services and products diﬃcult to obtain in thinly settled communities (theatre, variety in shopping malls, proximity to friends and relatives, etc.) and externalities are provided by the co-existence in one locale of productive activities requiring inputs of labor of similar skills. In the “new” economic geography, exempliﬁed by the recent book by Fujita, Krugman and Venables, increasing returns in production and transportation costs of the Samuelsonian “iceberg” variety between consumers and producers are important ingredients in the analysis of agglomeration. In their models, consumer behavior is explicitly modeled with the aid of the Dixit-Stiglitz utility function allowing a love of variety among commodities of the same general type, leading to a Chamberlinian form of monopolistic competition. At the outset of our paper we have taken liberties with this setting, concentrating instead on the costs of connecting fragments of a production process that can be outsourced, perhaps to other countries, and leaving ﬁnal consumers aside. The crucial issue is where are the increasing returns found - on the factory ﬂoor (i.e. within the production unit) or among the services required to link disparate fragments of the process. If increasing returns are found within each production block, which is our transformation of the Fujita, Krugman and Venables model to consider the outsourcing phenomenon, we argued that if transportation services are required to link two blocks, and if these services exhibit constant returns to scale (as in the iceberg model), larger scales of output will indeed tend to cause production fragments that are initially separated spatially, to agglomerate. However, we also provided an alternative scenario that leads to opposite conclusions, one based on the model we presented initially in 1990. A production process consists of a number of production blocks that can be fragmented and located in diﬀerent geographic regions of the same country, or can be outsourced to a variety of countries. The incentive to do so is provided by the diﬀerent skills or factor combinations required in various fragments and the variety of factor prices and/or factor skills available in diﬀerent regions or countries. Fragmentation allows a better “ﬁt” for each production block. But extra costs are involved - those of transportation, but also of ﬁnance, co-ordination, communication, etc, and we argue that it is in these service link activities that strong degrees of increasing returns and decreasing costs are to be found. To take extreme examples we as-

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sumed that production blocks exhibit constant returns to scale, while service link activities are purely of the ﬁxed cost variety, independent of scales of output. This diﬀerence in the location of the increasing returns activity is suﬃcient to lead to the result that eventually as output expands productive activity exhibits dis-agglomeration - a dispersal of productive activity to locations in which Ricardian and/or HeckscherOhlin diﬀerences among countries provide a better ﬁt for the separate fragments as the scale of production expands. Recent decades have witnessed profound productivity improvements in service links, whether of the transportation variety or in other service activities. The changes in communication costs have probably been the most signiﬁcant in lowering the service costs required to co-ordinate spatially separated production fragments. We have argued that such changes have encouraged dis-agglomeration both in the modiﬁed Fujita, Krugman and Venables scenario as well as in our model of international fragmentation. A melding of the two strands of argument concerning agglomeration was suggested in that international fragmentation of economic activity, promoted by larger scales of output and technological progress reducing the costs of service links, may lead to a subsequent agglomeration of fragments from diﬀerent industries, fragments that nonetheless require similar relative quantities and qualities of productive inputs. This can eventuate in a re-alignment of the location of production, with encouragement for further technological progress and externalities that serve, as well, to promote economic growth. Finally, we should note that in this, and our previous papers, we have remained relatively silent on such questions as: Will fragmentation take place within the ﬁrm or at arms-length in market transactions? Will large ﬁrms, possibly multinationals, dominate the process of international fragmentation while small ﬁrms are pushed away? These are legitimate and diﬃcult issues, and our silence reﬂects a lack of comparative advantage.11 Acknowledgements The authors wish to thank seminar participants in Helsinki and Osaka for valuable comments, as well as two anonymous referees. Bjarne Jensen has been very helpful in the revision of the manuscript. 11 For an informed view, consider Henry Wan, Jr. (2001). He uses the concepts of closed and open networks in discussing relations between subcontractors and assemblers. He argues persuasively, with many examples from Japan, Hong Kong, Korea and Taiwan, that no unique arrangement should be expected to emerge. Which particular organizational structure is appropriate and most eﬃcient to carry out fragmentation depends on the nature of an industry, the existence of information externalities, the speed of response to global transformation by individual ﬁrms and the economy as a whole.

International Trade and Agglomeration: An Alternative Framework

15

References Arndt, S, and Kierzkowski, H (eds.) (2001): Fragmentation: New Production and Trade Patterns in the World Economy. Oxford: Oxford University Press. Basu, S., and Fernald. J. (1997): “Returns to Scale in U.S. Production: Estimates and Implications.” Journal of Political Economy 105: 249-283. Bennarroch, M. (1997): “Returns to Scale in Canadian Manufacturing: An Interprovincial Comparison.” Canadian Journal of Economics 30: 1083-1103. Cheng, L., and Kierzkowski, H. (eds.) (2002): Globalization of Trade and Production in South-East Asia. New York: Kluwer Academic Press. Dixit, A. and Stiglitz, J. (1977): “Monopolistic Competition and Optimum Product Diversity.” American Economic Review 67: 297-308. Dumais, G,., Ellison, G., and Glaeser, E. (1997): “Geographic Concentration as a Dynamic Process.” NBER Working Paper 6270. Enke, S. (1951): “Equilibrium among Spatially Separated Markets: Solution by Electric Analogue.” Econometrica 19: 40-47. Feenstra, R., and Hanson, G. (1997): “Foreign Direct Investment and Relative Wages: Evidence from Mexico’s Maquiladoras.” Journal of International Economics 42: 371-94. Fujita, M., Krugman, P., and Venables, A. (1999): The Spatial Economy: Cities, Regions and International Trade. Cambridge (Mass.): MIT Press. Glaeser, E.L. (2000): “The New Economics of Urban and Regional Growth.” In The Oxford Handbook of Economic Geography, edited by G.L. Clark et. al. Oxford: Oxford University Press. Jones, R.W., and Kierzkowski, H. (1990): “The Role of Services in Production and International Trade: A Theoretical Framework.” In The Political Economy of International Trade, edited by R.W. Jones and Anne Krueger, ch. 3. Oxford: Blackwells. Jones, R.W., and Kierzkowski, H. (2001a): “A Framework for Fragmentation.” In Fragmentation: New Production and Trade Patterns in the World Economy, edited by S. Arndt and H. Kierzkowski. Oxford: Oxford University Press. Jones, R.W., and Kierzkowski, H. (2001b), “Horizontal Aspects of Vertical Fragmentation.” in Globalization of Trade and Production in South-East Asia, edited by L. Cheng and H. Kierzkowski. New York: Kluwer Academic Press. Krugman, P. (1991): “Increasing Returns and Economic Geography.” Journal of Political Economy 99: 483-99. Marshall, A. (1890): Principles of Economics. Book Four: The Agents of Production: Land, Labour, and Capital and Organization, Chapter 11, “Industrial Organization Continued. Production on a Large Scale” available on the Internet at: http://www.marxists.org/reference/subject /economics/marshall/bk4ch11.htm.

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Martin, R., and Sunley, P. (1996): “Paul Krugman’s Geographical Economics and its Implications for Regional Development Theory: A Critical Assessment.” Economic Geography 72: 259 - 292. Neary, J.P. (2001): “Of Hype and Hyperbolas: Introducing the New Economic Geography.” Journal of Economic Literature 39: 536-561. Ng, F., and Yeats, A. (2001): “Production Sharing in East Asia: Who Does What for Whom and Why?” In Globalization of Trade and Production in South-East Asia, edited by L. Cheng and H. Kierzkowski. New York: Kluwer Academic Press. Ohlin, B., Hesselborn, P.O., and Wijkman, P.M. (eds.) (1977): The International Allocation of Economic Activity. The Nobel Foundation. Porter, M.E. (2000): “Locations, Clusters, and Company Strategy.” In The Oxford Handbook of Economic Geography, edited by G.L. Clark et. al. Oxford: Oxford University Press. Samuelson, P.A. (1952): “Spatial Price Equilibrium and Linear Programming.” American Economic Review 42: 283-303. Samuelson, P.A. (1954): “The Transfer Problem and Transport Costs: The Terms of Trade when Impediments are Absent.” Economic Journal 64: 278-304. Sheppard, E. (2001): “How Economists Think: about Geography, for Example.” Journal of Economic Geography 1: 131-136. Sunley, P. (2001): “What’s behind the Models? ” in Critical Forum. Journal of Economic Geography 1: 136-139. Urban, D. (2001): “The Special Economy: One New Economic Geographer’s View.” Journal of Economic Geography 1: 146-152. Vernon, R. (1966): “International Investment and International Trade in the Product Cycle.” Quarterly Journal of Economics 80: 190-207. Wan, H., Jr. (2001): “Function vs. Form in the Fragmented Industrial Structure: Three Examples from Asia Paciﬁc Experience.” In Globalization of Trade and Production in South-East Asia, edited by L. Cheng and H. Kierzkowski. New York: Kluwer Academic Press. Yeats, A. (2001), “Just How Big is Global Production Sharing?” In Fragmentation: New Production and Trade Patterns in the World Economy, edited by S. Arndt and H. Kierzkowski. Oxford: Oxford University Press. Addresses of authors: – Ronald W. Jones, Department of Economics, University of Rochester, Rochester, New York 14627, USA (e-mail: [email protected]); – Henryk Kierzkowski, Graduate Institute of International Studies, HEI, 11A, Avenue de la Paix 1202, Geneva, Switzerland (e-mail: [email protected])

J. Econ. (2005) Suppl. 10: 17-56

General Equilibrium Dynamics of Multi-Sector Growth Models Bjarne S. Jensen and Mogens E. Larsen Received April 15, 2003; Revised version received August 9, 2004 c Springer-Verlag 2005

This paper analyzes Walrasian general equilibrium systems and calculates the static and dynamic solutions for competitive market equilibria. The Walrasian framework encompasses the basic multi-sector growth (MSG) models with neoclassical production technologies in N sectors (industries). The endogenous behavior of all the relative prices are analyzed in detail, as are sectorial allocations of the primary factors, labor and capital. Dynamic systems of Walrasian multi-sector economies and the family of solutions (time paths) for steady-state and persistent growth per capita are parametrically characterized. The technology parameters of the capital good industry are decisive for obtaining long-run per capita growth in closed (global) economies. Brief comments are oﬀered on the MSG literature, together with apects on the studies of industrial (structural) evolution and economic history. Keywords: pareto eﬃency, walrasian equilibria, factor accumulation JEL classiﬁcation: F11, F43, O40, O41.

1 Introduction Dynamics is concerned with calculating the motions of the widest variety of objects, and with deriving (computing) the implications (eﬀects) of these motions. Its basic principles and logical structure have long been a model for other scientiﬁc disciplines. The domain of dynamics in physics (mechanics) have been extended tremendously on both macroand microscopic scales, Pais (1986). In the discipline of economics, dynamics began in macroeconomics, in particular with business cycle and the basic one- and two-sector growth models. However, standard microeconomic (producer/consumer) theory is more naturally involved with decentralized mechanisms for resource allocation in static multisector modeling and general equilibrium dynamics. We may brieﬂy refer

18

Bjarne S. Jensen and Mogens E. Larsen

to passages from important theoretical and empirical contributions to multi-sectoral model building. Regarding the character of works in applied general equilibrium analysis (AGE-modelling), we quote Shoven & Whalley (1992, pp.1):1 “The central idea underlying this work is to convert the Walrasian general equilibrium structure (formalized in the 1950s by Arrow, Debreu, and others) from an abstract representation of an economy into realistic models of actual economies. Numerical, empirically based general equilibrium can then be used to evaluate concrete policy options by specifying production and demand parameters and incorporating data reﬂective of real economies” and “Most contemporary applied general models are numerical analogs of traditional two-sector general equilibrium models popularized by James Meade, Harry Johnson, Arnold Harberger, and others in the 1950s and 1960s. Earlier analytic work with these models has examined the distortionary eﬀects of taxes, tariﬀs, and other policies, along with functional incidence questions.” Computation of general equilibria usually involves solving systems of nonlinear equations. Thus, according to Judd (1998, p.147, p.3): “The Arrow-Debreu concept of general equilibrium reduces to ﬁnding a price vector at which excess demand is zero; it is the most famous nonlinear equation problem in economics” and “The computational general equilibrium (CGE) literature is the most mature computational literature in economics” Much of the numerical (algorithmic) methodology derives from the work of Scarf et al. (1967,1973). Numerical approaches give approximate solutions; many papers on numerical methods oﬀer little in the way of showing the qualitative dependence of the solutions upon critical parameter values. Obtaining analytical solutions in theorem-proof style is preferable wherever this is possible. Our emphasis in this paper will be to provide a conceptual framework that supports economic intuition and oﬀer a general and uniﬁed analytical approach to the mathematical procedures of obtaining the static, comparative static and dynamic general equilibrium solutions of 1

In all quotations throughout this paper Italics is ours.

General Equilibrium Dynamics of Multi-Sector Growth Models

19

an N-sector economy. We extend the methodology and mathematicaleconomic analysis of two-sector dynamics, Jensen (2003), to the multisectorial dynamics of temporal Pareto-eﬃcient labor and capital allocation/accumulation. An important contributor to multi-sector generalizations of various macrodynamic models was Jorgenson (1961), who extended the inputoutput methods of Leontief to dynamic input-output analysis. In the macro-planning and development literature, our topic of multi-sector economies had another important contributor in Johansen (1959, 1974, pp.1): “It is a well-known fact that the various sectors of an economy do not expand in the same proportion in a process of economic growth. The ﬂows of investment and new labor are not allocated proportionately to all production sectors. Existing quantities of capital and labor may be reallocated during the growth process. Terms of trade between the production sectors may change in a systematic way, and so on. Such considerations illustrate aspects of the economic growth process which we shall attempt to explain and analyze within the framework of a multi-sectoral growth model.” Regarding structural analysis and sectorial developments, the three categories of Colin Clark were dealing with the reallocation of the labour force over the groups of industries: Primary industries (agriculture, ﬁsheries and forestry), Secondary industries (manufacturing, handicraft, building and construction, Mining, Electric power production), Tertiary industries (wholesale and retail commerce, transport, ﬁnancial services and public administration), and globally observing, Clark (1951, p.365) that: “by careful generalization of available facts to be the most important concomitant of economic progress, namely the movement of working population from agriculture to manufacture and from manufacture to commerce and services.” Concomitant changes in the composition of demand, e.g., the budget share of food (Engel‘s law, 1857) is conﬁrmed by all surveys, Houthakker (1957). Controversies related to “convergence” and “balanced growth” models as being incompatible with structural change and the process of economic development are given renewed attention in both the empirical and theoretical growth literature, Pasinetti (1981), Islam (1995), Echevarria (1997), Laitner (2000), Kongsamut, Rebelo, Xie (2001), Meckl (2002). Patterns of sectorial industrial growth will here be obtained and shortly discussed.

20

Bjarne S. Jensen and Mogens E. Larsen

As we consider the relationship between resources, technology, and economic evolution, we ﬁnally quote the historian, A. P. Usher (1954, p.1): “Economic history is concerned with the description and the analysis of the mutual transformations taking place between human societies and their environment. The study of costs and prices is important, and the institutional structure of organized social life demands careful attention, but the basic problems of economic history lie in the ﬁeld of the management of resources. - The quantitative analysis of economic activities requires study of the processes and accomplishments of the system of production in physical units as well as in value units.” This paper deals with dynamic foundations of MSG models for closed (global) economies. We explain well-known empirical facts by modeling in terms of basic microeconomic principles. Section 2 presents an analytical framework with concepts, deﬁnitions, and various micro and macro economic equivalence relations. Section 3 studies the relationships between factor prices and relative commodity prices. In section 4, we analyze and derive GDP expenditure shares from some speciﬁc parameterizations of utility functions. Section 5 uses a proper NIPA version of Walras’s law to obtain the Walrasian equilibrium of the multi-sector economy, and we derive the timeless (static and comparative static) competitive general equilibrium solutions for all the variables as distinct composite functions of the factor endowments. In section 6, we analyze the dynamic systems and alternative evolutions of multisector general equilibria. Section 7 gives various asymptotic sectorial growth rates of persistently growing multi-sector economies. Our ﬁnal comments are found in section 8.

2 Analytical Framework for Multi-Sector (MS) Economies 2.1 The Supply Side, Technology and Eﬃcient Factor Allocation Consider an economy consisting of N industries (sectors), and let sector 1 be a capital good industry. Sector technologies, Fi (Li , Ki ), are described by nonnegative smooth concave homogeneous production functions with constant returns to scale in labor and capital, i = 1, · · · , N , Yi = Fi (Li , Ki ) = Li Fi (1, ki ) ≡ Li fi (ki ) ≡ Li yi , Li = 0, Fi (0, 0) = 0, (1)

General Equilibrium Dynamics of Multi-Sector Growth Models

21

where the function fi (ki ), is strictly concave and monotonically increasing in the capital-labor ratio ki ∈ [0, ∞[, i.e., fi has the properties ∀ki > 0 : fi (ki ) =

d2 fi (ki ) dfi (ki ) < 0, > 0, fi (ki ) = dki2 dki

lim fi (ki ) ≡ b¯i ≤ ∞,

ki →0

(2) lim fi (ki ) ≡ bi ≥ 0, fi (ki ) ∈ Ji ≡ bi , b¯i .

ki →∞

(3) Let us note that if our Fi is deﬁned and continuous on the axes, then Fi (0, 1) ≥ 0, Fi (1, 0) ≡ fi (0) ≥ 0, Fi (0, Ki ) = Ki Fi (0, 1), Fi (Li , 0) = Li Fi (1, 0) = Li fi (0),

(4) (5)

where positivity in (4) says that a factor is non-essential. Note also, ∂Fi (1, 0) = Fi (1, 0). ∂Li (6) The sectorial output elasticities, Li , Ki , i , with respect to marginal and proportional factor variation are, cf. (1), lim fi (ki ) =

ki →0

∂Fi (1, 0), ∂Ki

lim fi (ki ) = Fi (0, 1),

ki →∞

M PLi ki fi (ki ) ∂Yi Li ∂ ln Yi > 0, ki = 0, (7) =1− = ≡ fi (ki ) APLi ∂Li Yi ∂ ln Li M PKi d ln yi ki fi (ki ) ∂Yi Ki ∂ ln Yi > 0, (8) = = = ≡ ≡ d ln ki fi (ki ) APKi ∂Ki Yi ∂ ln Ki ≡ Li + Ki = 1. (9)

Li ≡ Ki i

The factor endowments, total labor force (L) and the total capital stock (K), are inelastically supplied and are fully employed (utilized), i.e. L= K=

N

N i=1 Li /L ≡ i=1 λLi = 1, λLi N N i=1 Ki /K ≡ i=1 λKi ≡ 1, N N i=1 λLi ki ≡ i=1 li ki ,

i=1 Li , N i=1 Ki ,

K/L ≡ k ≡

N

= li ,

(10) (11) (12)

where λLi , λKi , (10-11) are the factor allocation fractions. At any point of the isoquants (1), the marginal rates of technical substitution (MRS), ωi (ki ) are, by (2), positive monotonic functions, ωi = ωi (ki ) =

L M PLi fi (ki ) − ki = i ki > 0 = Ki fi (ki ) M PKi

∀ki > 0,

(13)

22

Bjarne S. Jensen and Mogens E. Larsen

and the substitution elasticities, σi between labor and capital is the proportionate change in the ratio (Ki /Li ) of inputs divided by the proportionate change in the ratio (ωi ) of the marginal products of inputs, cf. (13), (7-8), d ln APKi d ln APLi dki Li dki ωi d ln ki > 0. (14) = = = ≡ d ln M PKi d ln M PLi dωi Ki dωi ki d ln ωi

σi ≡

The general relation between the sectorial factor output elasticities, (7-8), and the sectorial substitution elasticities, σi , is: K (1 − σi ) dLi ki L (σi − 1) d ln Li dKi ki d ln Ki . = i ≡ , = i ≡ σi dki Li d ln ki σi dki Ki d ln ki (15) Free factor mobility between the multiple industries and also eﬃcient factor allocation impose the common MRS condition, cf. (13), ω = ωi = ωi (ki ) (i = 1, ..., N ).

(16)

For the variables ki to satisfy (16), it is, beyond (2), further required that the intersection of the sectorial range for ωi (ki ) is not empty, ¯ i ] ⊆ R+ , ωi (ki ) ∈ Ωi = [ω i , ω

¯ ] = ∅. ω ∈ Ω ≡ ∩ Ωi = [ω, ω

(17)

2.2 Eﬃcient Factor Allocation, Costs, and Relative Prices All industries are assumed to operate under perfect competition (zero excess proﬁt); absolute (money) input (factor) prices (w, r) are the same in both industries; and absolute (money) output (product, commodity) prices (Pi ) represent unit cost. Thus, in each sector we have the competitive producer equilibrium equations, w = Pi · M PLi , r = Pi · M PKi ; ω = w/r, Pi = 0, (18) Pi Yi = wLi + rKi , Li = wLi /Pi Yi , Ki = rKi /Pi Yi , (19) Pi = (w + rki )/yi , Pi = Li w ˆ + K i rˆ, (Pi ≡ dPi /Pi ). (20) ˆ and rˆ. Thus the relative change, Pi , is a convex combination, (9), of w Next, (18) gives any relative commodity price Pi /Pj as pij ≡

yj Lj M PLj M PKj fj (kj ) − kj fj (kj ) fj (kj ) Pi . = = = = = yi Li M PLi fi (ki ) − ki fi (ki ) fi (ki ) M PKi Pj (21)

General Equilibrium Dynamics of Multi-Sector Growth Models

23

The connection between relative factor (service) prices and relative commodity prices follows from (16, 18, 21), −1 M PKj [kj (ω)] fj [kj (ω)] w fi [ki (ω)] Pi , ω= . = = pij (ω) = (ω) = r fj [kj (ω)] fi [ki (ω)] M PKi [ki (ω)] Pj (22) With a common wage-rental ratio, (16), in all sectors it is clear that, Li (ω) Lj (ω) ⇔ Kj (ω) Ki (ω) ⇔ kj (ω) ki (ω).

(23)

Average and marginal productivities were objects of pioneering theoretical and empirical studies by von Th¨ unen. In Table 1 below, the factor K is non-essential, cf. (4); accordingly, the substitution elasticity is larger than one. Hence L , (7), (15), (18), (19), is here falling with mechanization, K/L. Evidently, with σ > 1, APL increases more than M PL , (14). This table may serve as a relevant and simple illustration of key sectorial productivity growth numbers. Table 1. Von Th¨unen´s Productivity Calculations. ω M PK k· K/L APL APK M PK M PL 0 110 − − 110 − − 1 150 150 40 110 2.75 40 2 186 93 36 114 3.17 72 3 218.4 72.8 32.4 121.2 3.74 97.2

εL 1.00 0.73 0.61 0.55

Source: von Th¨ unen (1850 [1930, p. 507]); Brems (1986, p. 86).

2.3 Macro-Equivalence Relations of Supply-Demand in MS Economies Gross domestic product (GDP), national income, Y , is the total of sectoral producer revenues [monetary value of sector outputs, (1)] N N Yi = Lyi li , Y ≡ i=1 Pi Yi = L (24) i=1 Pi yi li ≡ Ly, and is equivalent with competition (18) and (19) to total factor income, Y = wL + rK = L(w + rk) = L(ω + k)Pi fi [ki ] = Ly.

(25)

Hence the factor income distribution shares δK + δL = 1, become δK ≡ rK/Y = rk/y, δL ≡ wL/Y = w/y; δK = k/(ω + k), δK /δL = k/ω. (26)

24

Bjarne S. Jensen and Mogens E. Larsen

The “ﬁnal demand” decomposition of GDP (Y) into the aggregate expenditures on investment (I) and consumption (C) is I = P1 Y1 ,

C=

N

i=2

Pi Yi ,

Y = C + I,

(27)

and the composition of GDP (24) expenditure shares, si is si = Pi Yi /Y,

N

i=1 si

≡

N

i=1

Pi Yi /Y = 1.

(28)

Budget studies and consumption (demand) theory mostly normalize the budget shares with the total expenditure (“income”) constraint, C, i.e., the expenditure shares, ei , as N

e + e¯ = 1. (29) Evidently, the connection between the GDP shares (28) and (29) is ei = Pi Yi /C,

i=2 ei

= 1; e =

σi <1 ei ,

si = (1 − s1 )ei ,

e¯ =

σi >1 ei ,

i = 2, ..., N.

(30)

For later purposes, we introduce the notation, cf. (29), s = (1 − s1 )e + s1 , σ1 > 1 : s = (1 − s1 )e, σ1 < 1 : s = (1 − s1 )e + s1 , s = (1 − s1 )e,

s + s = 1. (31)

Lemma 1. The macro factor income shares δL , δK , (26), are GDP expenditure-weighted combinations of the sectorial factor income (cost) shares, Li , K i , δL =

N

i=1 si Li ,

δK =

N

i=1 si K i ,

δK + δL = 1.

(32)

The factor allocation fractions (10) and (11) are obtained by Li /L = λLi = li = si Li /δL ,

Ki /K = λKi = si Ki /δK .

(33)

The total factor endowment ratio K/L satisﬁes the identity [cf.(12),(26),(32)]: N N ωδK K N = ω i=1 si Ki = k = i=1 λLi ki = i=1 si Li , (34) δL L

which is a convenient representation of Walras’s law (identity).

General Equilibrium Dynamics of Multi-Sector Growth Models

25

Proof: By deﬁnition, we have δL = wL/Y = [wL1 + wL2 + · · · wLN ] /Y, δK = rK/Y = [rK1 + rK2 + · · · rKN ] /Y.

(35) (36)

From (19) and (28), we get wLi = Li Pi Yi = si Li Y,

rKi = Ki Pi Yi = si Ki Y.

(37)

Hence, by (35), (36) and (37), we obtain (32). Next, as stated in (33), si Li si Li Y wLi Li , = = = δL δL Y wL L si Ki si Ki Y rKi Ki . = = = = δK δK Y rK K

(38)

λLi = λKi

////

(39)

3 Relative Prices in Economies with CES Technologies The general CES forms of Fi (Li , Ki ), (1), γi > 0, 0 < ai < 1 and σi > 0 are i Yi = Fi (Li , Ki ) = γi L1−a Kiai = Li γi kiai , ≡ Li fi (ki ), i i σ σ−1

σi −1 σi −1 i σi σi + ai K i Yi = Fi (Li , Ki ) = γi (1 − ai )Li

(40) (41)

σ /(σi −1) (σ −1)/σi i (42) = Li γi (1 − ai ) + ai ki i ≡ Li fi (ki ), 1/(σi −1) −(σ −1)/σi fi (ki ) = γi ai kiai −1 , fi (ki ) = γi ai ai + (1 − ai )ki i .

(43) By evaluating equations (40)-(43), the limits of fi (ki ) and fi (ki ) can σ /(σ −1) be written as follows σi ≷ 1 ⇒ ai i i ≷1 : ⎧ σi ⎪ lim fi (ki ) = γi (1 − ai ) σi −1 , ⎨ lim fi (ki ) = 0, k →0 ki →∞ σi (44) σi < 1 i σi −1 ⎪ ⎩ lim fi (ki ) = γi ai , lim fi (ki ) = 0, ki →0

ki →∞

⎧ σi ⎪ ⎨ lim fi (ki ) = γi (1 − ai ) σi −1 , lim fi (ki ) = ∞, ki →∞ k →0 σi (45) σi > 1 i σ −1 ⎪ ⎩ lim fi (ki ) = ∞, lim fi (ki ) = γi ai i . ki →0

ki →∞

26

Bjarne S. Jensen and Mogens E. Larsen

For the CES technologies, the monotonic relations between marginal rates of substitution, factor proportions, and output elasticities are [cf. (41-43)] σ

1 − ai i 1 1 − ai 1/σi σi (46) ki , ki = [ωi ] , ci = ωi = ai ci ai ,

i −1 ci ω 1−σi 1 1 − ai 1−σ σi . (47) , = = ki Ki = 1 + L i 1 + ci ω 1−σi 1 + ci ω 1−σi ai With multi-sector models and CES technologies, it is apparent, cf. (46), that sectorial factor ratio (“intensity”) reversals can only be avoided if and only if σi = σj and ai = aj . Hence, with σi = σj , the reversal point ¯ ω ¯ ω is, (ki , ωi ) = (kj , ωj ) = (k, ¯ ), [ai = aj = a : (k, ¯ ) = (1, 1−a a )] :

ai (1 − aj ) k¯ = aj (1 − ai )

i σj σσ−σ j

i

=

cjσi σ

ci j

1 σ −σ

j

,

i

cj ω ¯= ci

1 σ −σ

j

i

(48)

.

3.1 The Correspondence of Product and Factor Prices with CES The exact form of the function (22) needs particular attention. With (43) and (46), the relative commodity prices (comparative costs) (22) become, with σi = 1, σi = 1, and σi = σ, respectively, a

pij (ω) =

γj aj j (1 − aj )1−aj aj −ai fj [kj (ω)] γj aj kj (ω)aj −1 ω , = = γi aai i (1 − ai )1−ai γi ai ki (ω)ai −1 fi [ki (ω)]

(49) 1/(σ −1) j fj [kj (ω)] γj aj aj + (1 − aj )kj (ω)−(σj −i)/σj (50) = pij (ω) = 1/(σi −1) fi [ki (ω)] γi ai ai + (1 − ai )ki (ω)−(σi −1)/σi 1/(σj −1) σ /(σ −1) γj aj j j 1 + cj ω 1−σj , (51) = σ /(σ −1) 1/(σi −1) γi a i i i [1 + ci ω 1−σi ] σ

1/(σ−1)

σ fj [kj (ω)] 1 − ai 1 + cj ω 1−σ aj γj (52) c = = pij (ω) = i ai 1 + ci ω 1−σ , ai γi fi [ki (ω)] . The elasticity of the functions (22), (49), (50), (51) and (52) are generally interlinked by the composite rule: E [pij , ω] = E M PK j , kj E(kj , ω) − E [M PK i , ki ] E(ki , ω) (53) = (−Lj /σj )σj − (−Li /σi )σi = Li − Lj = K j − K i . (54)

General Equilibrium Dynamics of Multi-Sector Growth Models

27

Evidently, pij (ω) is always inelastic, as (54) is numerically less than unity; but this is not directly seen from the explicit CES expressions (49), 51) and (52). The Stolper-Samuelson theorem and the “magniﬁcation eﬀects” of price changes [cf. Jones (1965)] are implications of the elasticity between the factor and commodity prices (FPCP elasticity) (53)-(54), which can be rewritten as

Pi − Pj

w ˆ − rˆ = Li − Lj = K j − K i .

(55)

ˆ and rˆ, cf. (20), then Since Pi and Pj both lie between w | Pi − Pj w ˆ − rˆ | < 1.

(56)

Combining (54, 55), (56), with (23), we immediately obtain the following inequalities (“magniﬁcation relations”): ki (ω) < kj (ω) : ki (ω) > kj (ω) :

Pˆi > Pˆj : w ˆ > Pi > Pj > rˆ, ˆ Pˆj > Pˆi : rˆ > Pj > Pi > w,

(57)

Pˆi > Pˆj : rˆ > Pi > Pj > w, ˆ ˆ > Pj > Pi > rˆ. Pˆj > Pˆi : w

(58)

Since the CES marginal rate of substitution ωi in (46) always has the limit values zero and inﬁnity, we need, for precise geometry and intuition, to know the limits of the relative prices pij (ω), (51), for ω going to zero and inﬁnity. To this end, let σ /(σ −1)

p∗ij

j j γj aj , ≡ γi aσi /(σi −1)

i

p∗∗ ij ≡

γj (1 − aj )σj /(σj −1) . γi (1 − ai )σi /(σi −1)

(59)

With σi = σj , (51), (48), any relative price has a unique reversal price ratio: j 1 σ1−σ c j −σi σj −1 1 + cj cji ω) = p¯ij = pij (¯ 1 ;

σi i σ1−σ cj j −σi σi −1 σi −1 1 + ci ci γi ai

σj σ −1

γj aj j

ai = aj ,

p¯ij =

γj . γi (60)

28

Bjarne S. Jensen and Mogens E. Larsen

Proposition 1. The graphs of the relative prices, pij (ω), (51) – the CES factor-price-commodity-price (FPCP) correspondence – have limits, classiﬁed by σi , as follows: σi < 1, σj < 1 : lim pij = p∗ij , ω→0

σi > 1, σj > 1 : lim pij = ω→0

lim pij = p∗∗ ij ,

(61)

p∗ij ,

(62)

ω→∞

p∗∗ ij ,

lim pij =

ω→∞

σi > 1, σj < 1 : lim pij = 0,

lim pij = 0,

(63)

lim pij = ∞.

(64)

ω→∞

ω→0

σi < 1, σj > 1 : lim pij = ∞,

ω→∞

ω→0

ω ), (60), is always a maximum (iﬀ The reversal price ratio, p¯ij ≡ pij (¯ σi > σj ) or a minimum (iﬀ σj > σi ) [cf. Fig. 1]. For the substitution elasticities (61)-(63), the range of pij (ω), (51), is bounded. With γi = γj , and both substitutions elasticities, either small (61) or large (62), the range of pij (ω) becomes a narrow interval, and there will be only small diﬀerences between the values of p∗ij and p∗∗ ij , (59), if ai , aj , have similar size [cf. Table 4]. Iﬀ σi = σj = 1, the functions, pij (ω), (52), are always monotonic, bounded, and increasing between p∗ij and p∗∗ ij , iﬀ aj > ai . Only the CD relative prices pij (ω), (49), are monotonic and unbounded, cf. Fig. 1. Proof: First, results (61)-(62). With σi < 1 and σj < 1, the brackets in (51) converge to one at ω = 0, which establishes the limiting p = p∗ . For large ω, the numerator and denominator in (51) are approximately proportional to ω −1 with coeﬃcients equal to the numerator and the denominator of p∗∗ , (59). Similar arguments as above establish (62). Second, results (63)-(64). With σi > 1 and σj < 1, the numerator in (51) converges to a constant and the denominator converges to inﬁnity at ω = 0, which establish the limiting value of p = 0. For large ω, the numerator of (51) converges to zero, whereas the denominator converges to a non-zero constant. Hence the limiting value of p is zero. Similar arguments as above establish (64). //// The concrete values of, p∗ij , p∗∗ ¯ij can be calculated for any size ij and p of the CES parameters; however, for a clear quantitative understanding of the relevant interval for any relative price, we examine the quotients [cf. (59), (60)] p∗∗ ij q∗ = ∗ pij q¯ =

p¯ij p∗ij

σj

[(1 − aj )/aj ] σj −1

1

cj σj −1

q¯ p¯ij (65) ∗∗ = q ∗ , p ij ci [(1 − ai )/ai ] σ 1−1 σ 1−1

1−σj 1−σi i j σj −σi σj −σi (66) 1 + ci (cj /ci ) = 1 + cj (cj /ci ) =

σi σi −1

=

1 σi −1

,

,

σj −σi

ai = aj = a : q¯ = a (σi −1)(σj −1) .

(67)

General Equilibrium Dynamics of Multi-Sector Growth Models

29

If aj = ai = 12 , then q ∗ = 1 and q¯ form the interval of relative prices. Although the numerical value of q ∗ is sensitive to its many parameter combinations, the dominating inﬂuences on the deviation of this quotient from q ∗ = 1 are the extent to which both, aj , ai , diﬀer from the critical value 12 , and the extent to which both, σi , σj , diﬀer from the critical value 1. Regarding the size of the quotient q ∗ , let δl and dl be: σi −σj

σl 1 − al (σi −1)(σj −1) (1 − aj )/aj σl −1 (l = i, j). , δl = Aji , dl = Aji = al (1 − ai )/ai (68) Lemma 2. Exact decompositions of the quotient q ∗ , (65), are: q ∗ = δj di = δi dj .

(69)

Proof: With q ∗ from (65) and dl from (68), we get (after reductions) σj σ −1

q ∗ = Ajij

σi σ −1

di = Ajii

dj ,

(70)

which is (69). From (65)-(68) it follows that

////

limσi ,σj →∞ q ∗ = Aji ,

limσi ,σj →0 q ∗ = 1.

(71)

We illustrate Lemma 2 in Tables 2-4. Table 2

Table 3

ai ≥ aj ai ≤ aj Aji ≥ 1 Aji ≤ 1 σl > 1 δ l ≥ 1 δ l ≤ 1 σl < 1 δ l ≤ 1 δ l ≥ 1

al ≤ 12 al ≥ 12 σi ≤ σj dl ≥ 1 dl ≤ 1 σi ≥ σj dl ≤ 1 dl ≥ 1

Table 4. Numerical illustrations of benchmark cases (γi = γj = 1). ai 0.3 0.3 0.8 0.3 0.3 0.4 0.3 0.3 0.3

aj 0.25 0.25 0.25 0.40 0.30 0.30 0.25 0.25 0.25

σi 0.1 0.4 0.3 0.6 0.6 0.6 1.6 2.0 4.0

σj 0.1 0.3 0.2 0.7 0.7 0.7 1.5 1.9 4.0

Aji 1.29 1.29 12.0 0.64 1.00 1.56 1.29 1.29 1.29

δi 0.97 0.85 0.34 1.94 1.00 0.52 1.95 1.65 1.40

δj 0.97 0.90 0.54 2.80 1.00 0.36 2.13 1.70 1.40

di 1.00 1.22 0.78 0.49 0.49 0.71 1.33 1.10 1.00

dj 1.00 1.30 1.22 0.71 0.49 0.49 1.44 1.13 1.00

q∗ 0.97 1.10 0.42 1.38 0.49 0.25 2.82 1.87 1.40

∗ pij 1.02 0.81 1.29 1.39 2.73 4.20 0.39 0.60 0.78

∗∗ pij 0.99 0.89 0.54 1.93 1.35 1.07 1.09 1.11 1.10

p¯ij − 1.01 1.29 0.91 1.00 0.92 1.11 1.11 −

q¯ − 1.24 1.00 0.65 0.37 0.22 2.86 1.87 −

Case 1.1, si = sj = 1, aj > ai

Case 1.2, sj = si = 1, aj < ai

w/r

w/r

wi

wj

wj Pi/Pj

wi k

Case 1.3, si = sj < 1, aj > ai

w/r

Pi/Pj

Pi/Pj

k

Case 1.4, si = sj < 1, aj < ai

wi

w/r

wj

wj

k

pij** pij*

Case 1.5, si = sj > 1, aj > ai

Pi/Pj

pij*

wi

k

pij**

Case 1.6, sj = si > 1, aj < ai

w/r

w/r

wi

wj wi

wj Pi/Pj

k

pij* pij**

Case 1.7.1, sj < si < 1, aj £ ai > ½, pij* > pij** wj

w/r

Pi/Pj

Case 1.7.2, si < sj < 1, ½ < aj ³ ai, pij* < pij** w/r

wi

w Pi/Pj

pij

pij* pij**

k w/r

wj

Pi/Pj

pij** pij* pij

k

k w/r

Pi/Pj

pij* pij**

pij

k

Pi/Pj

pij* pij**

pij

k

k

pij

Pi/Pj

pij** pij*

pij

Case 1.10, si < 1, sj > 1

wj

w Pi/Pj

wj k

k w/r

wi

w

wi

k

wi

w

wi

w/r

k

k w/r

wj

w

Case 1.9, si > 1, sj < 1

wj

Case 1.8.4, 1 < si < sj, aj £ ai > ½, pij* < pij**

w/r

pij*pij**

wi

w k

k

k w/r

wi

Case 1.8.3, 1 < sj < si, aj ³ ai < ½, pij* > pij**

pij

wj

Case 1.8.2, 1 < si < sj, aj ³ ai < ½, pij* > pij**

wj

w

Pi/Pj

wi

w

pij**pij*

pij pij** pij*

k

k w/r

Case 1.8.1, 1 < sj < si, ai ³ aj < ½, pij* < pij**

Pi/Pj

wj

Case 1.7.4, si < sj < 1, ½ < ai ³ aj, pij* > pij**

wi

w

pij

wi

w k

Case 1.7.3, sj < si < 1, aj ³ ai < ½, pij* < pij**

Pi/Pj

k

pij** pij*

k

Pi/Pj

pij

wj

k

Fig. 1. FPCP correspondence p ij (ω), [cf. (49), (51) and (52)].

k

General Equilibrium Dynamics of Multi-Sector Growth Models

31

Due to (9), the relationship between relative product (cost) prices and factor prices is not aﬀected by factor endowments, or by any demand or any taxation/tariﬀ system, speciﬁed in CGE models. Thus, as an invariance, this FPCP pij (ω), (51), is implicitly embedded in several applied CGE models (with (9) and CES technologies), and hence it can also be retrieved explicitly from (or check) presentations of the complete general equilibrium solution. The well-known CGE calculations in Table 5 use CES parameters: γi = 1.5, γj = 2.0, ai = 0.4, aj = 0.3, σi = 2.0, σj = 0.50

(72)

i.e., refers to Case 1.9, Fig. 1, with maximum here, p¯ij = 1.35 , cf. (60). Table 5. Computational General Equilibrium solutions,2 and the FPCP correspondence, pij (ω), (51). w 1.00 1.00 1.00 1.00 1.00

r 1.373 1.426 1.393 1.372 1.438

ω = w/r 0.728 0.701 0.718 0.729 0.695

pij (ω) 1.280 1.271 1.277 1.280 1.268

Pi 1.399 1.412 1.404 1.399 1.415

Pj 1.093 1.111 1.100 1.092 1.115

Pi /Pj 1.280 1.271 1.276 1.281 1.269

4 Preferences, Demand and Expenditure Shares of GDP 4.1 Consumption and Saving On the demand side of the economy, the actual division of national income between saving and consumption is our ﬁrst problem, posing major issues of long theoretical and empirical standing. However, attention here will only be given - as in NIPA accounting - to the accumulation of new productive capital (tangible assets), excluding intangibles such as all services, education (human capital), and portfolio & wealth evaluations (“capital gains”) from entering NIPA saving accounts. Hence only the newly produced ﬁnal goods of a few manufacturing, construction and building industries will enter the NIPA saving/investment share of GDP. Thus, with only one capital good (“equipment”) industry, we have Y = C + S, s = S/Y = I/Y = P1 Y1 /Y = s1 . (73) Per capita saving/consumption, in any numeraire, are denoted by sL /Pi , cL /Pi , s = S/Y = (S/L)/(Y /L) = sL /y = (sL /Pi )/(y/Pi ), 2

Shoven and Whalley (1992), Tables 3.1 - 3.7.

y − sL = cL . (74)

32

Bjarne S. Jensen and Mogens E. Larsen

Optimum paths for savings and consumption by a representative agent may be described in terms of the intertemporal utility functional: ∞

cL (t) e−R[u(t)] dt, (75) U= u Pi 0 where R[u(t)] is an accumulated rate of time preference by which future utility is discounted – and which summarizes the preference structure of the agent regarding the time proﬁle of the continuous utility stream from present and future consumption, see Uzawa (1969, p. 630), (1968, p. 488). According to Uzawa (1969, p. 634), (1973, p. 58), if the intertemporal preference orderings are homothetic, then the optimal per capita saving (sL /Pi ) is separable and linear with respect income, (y/Pi ), i.e., sL /Pi = sL (r, w, y) = sL (r/Pi , w/Pi , y/Pi ), sL /Pi = sL (r/Pi , w/Pi , y/Pi ) = s∗L (r/Pi , w/Pi ) · (y/Pi ).

(76) (77)

Thus with (77) and (74) the overall saving rate (s) is solely determined by the “real” factor prices, and such that the saving rate (s) increases with the real rental rate and decreases in the real wages, i.e., s = s∗L (r/Pi , w/Pi );

∂s∗L /∂(r/Pi ) > 0,

∂s∗L /∂(w/Pi ) < 0.

(78)

A further reduction in the arguments of saving function (78) may occur in a general equilibrium context with Pareto optimality [(cf. (16), (18) and (22)] (79) s = s∗L (r/Pi )(ω), (w/Pi )(ω) = s∗∗ L (ω), which might allow the budget share (s) to be also formally handled by homothetic utility functions, applied to genuine consumer goods below. In the tradition of the B¨ ohm-Bawerk & Fisherian theory of time preference, Koopmans (1960), Tinbergen (1960), Uzawa (1973, p. 59), (1968, p. 494), (1969, p. 630, 637), the optimal saving rate is essentially dependent on (increasing with) the expected real rate of interest, ier , with its level being equal to the marginal rate of time preference, ρ[u(t)] , i.e., in short, s = s(ier ), (r/P1 )e = ier = ρ[u(t)]. (80) A serious problem is, however, that the saving functions (76)-(80) have not so far obtained generally accepted functional forms, and even less, been examined for critical values of fundamental parameters. Furthermore, the saving/investment share of GDP is not just a matter of optimizing consumer preferences (temporal/intertemporal). Actually, it needs to be based on joint considerations of the producer-consumer agent.

General Equilibrium Dynamics of Multi-Sector Growth Models

33

Hence, with some empirical support, we adopt a provisional assumption of mathematically treating the saving rate as a parametric (“ﬁxed”) constant: s = s1 = constant (parametric variation).

(81)

This assumption does not preclude seeing positive eﬀects of relatively declining capital good prices on investment (capital stock accumulation), cf. (73). As to the observed long-run constancy of the gross private (household and business combined ) saving rate, see Denison (1958, p.267), David & Scadding (1974, p.238), Tobin (1980). The constancy of the total private saving rate (81) may be interpreted, Tobin (1980, p.65), as an extension of the Modigliani-Miller theorem beyond ﬁnance to real capital accumulation.

4.2 Expenditure Systems and CES-Class Utility Preferences The purpose of budget (expenditure) systems is to describe mathematically how in a certain period a given money amount of total consumption expenditures is allocated to item-speciﬁc expenditure categories. The key elements determining this budget allocation are, as usual, assumed to be consumer preferences, prices of consumer goods & services, and the given size of the total budget (expenditures). Rather than giving Marshallian demand or Hicks/Slutsky (compensated) demand functions for various preference functions, we need explicit analytical expressions of the budget shares, ei (29), for here actually solving the static and dynamic general equilibrium systems. We shall make use of a few benchmark speciﬁcations of preference (utility) functions - direct, U (.), and indirect utility, V (.) - cf. MassColell et al. (1995, p. 56), Deaton & Muelbauer (1992, p. 38). Among many other references, we refer to, Silberberg & Suen (2001), Chung (1994), Wold (1952). Introducing the notation of expenditure (“income”) and price elasticities as, ∂ ln Yi /∂ ln C ≡ E(Yi , C), ∂ ln Yi /∂ ln Pj ≡ E(Yi , Pj ), then demand (expenditure) systems must satisfy four basic, well-known, Frisch (1959, p.180), restrictions - which also must be met by our general equilibrium solutions: N Homogeneity: (82) j=2 E(Yi , Pj ) + E(Yi , C) = 0, N (83) Engel Aggregation : j=2 ej · E(Yj , C) = 1, N (84) Cournot Aggregation : j=2 ej · E(Yj , Pi ) = − ei < 0, Symmetry: E(Yi , Pj )/ej + E(Yi , C) = E(Yj , Pi )/ei + E(Yj , C). (85)

34

Bjarne S. Jensen and Mogens E. Larsen

Incidentally, note that any demand (expenditure) functions with constant non-unitary price and income elasticities cannot satisfy (83), except for a narrow range of the price and income variables involved, Wold (1952, p.106). The CES form of the direct/indirect utility functions is the only functional form with the property of self-duality, i.e., the dual can be expressed exactly with same parameters; see, Samuelson (1965), Houthakker (1965). We have [Silberberg & Suen (2001, p. 359), Chung (1994, p. 58), N Shoven & Whalley (1992, p. 45, 96)]: αi > 0, i=2 αi = 1,

CES: U (Y2 , ..., YN ) = γu

N

i=2

αi Yi

σu −1 σu

σ σu−1 u

(86)

,

1 1 N σu −1 σu −1 σu α [C/P ] j j=2 j γu 1 1 N σu σu −1 σu −1 C/Pi , α p = ij j=2 j γu ⎤−1 ⎡ σu N

α αiσu Pi1−σu Pi Yi j σu −1 ⎦ pij =⎣ = N ei = σu 1−σu α C i α P j j=2 j j=2 V (P2 , ·, PN , C) =

(87)

(88)

,

−1

σu N αj σu −1 pij (i = 2, ..., N ). ei = ei (pi2 , ..., piN ) = 1 + αi j=2,j=i

(89) Linking the relative consumer prices of ei , (89), to cost prices, (22) and (51), we obtain

−1 N ei (ω) = 1 + (αj /αi )σu pij (ω)σu −1

i > 1,

,

j=2,j=i

0 < ei (ω) < 1. (90)

Lemma 3. The limiting consumer expenditure shares – with CES technologies (41–43) and CES utility functions (86) – are given by σu = 1 : ∀i σi < 1 : lim ei (ω) = e∗i ,

lim ei (ω) = e∗∗ i , (91)

ω→0

ω→∞

ω→0

ω→∞

σu = 1 : ∀i σi > 1 : lim ei (ω) = e∗∗ i ,

lim ei (ω) = e∗i ,

(92)

General Equilibrium Dynamics of Multi-Sector Growth Models

35

where e∗i

N

= 1+

(αj /αi )σu p∗ij σu −1

j=2,j=i

−1

N

∗ i=2 ei

i > 1, ,

−1 N σu ∗∗ σu −1 e∗∗ = 1 + (α /α ) p j i i ij

i > 1,

N

∗∗ i=2 ei

,

j=2,j=i

= 1,

(93)

= 1, (94)

and pij∗ , pij∗∗ are given by (59), cf. Proposition 1. When some of the industries have CES substitution elasticities larger and smaller than one, the limiting expenditure shares become: σu < 1, σi > 1 > σj : lim ei (ω) = 0; lim ej (ω) = e∗j , ω→0,∞

ω→0

σu > 1, σi > 1 > σj : lim ei (ω) = e¯∗∗ ¯∗i , i , lim ei (ω) = e ω→∞

ω→0

lim ej (ω) = e∗∗ j , (95)

ω→∞

lim ej (ω) = 0, (96)

ω→0,∞

where

e¯∗i = 1 +

e¯∗∗ = 1+ i

∗

(αj /αi )σu pij∗ σu −1

(αj /αi )

j=2,j=i,σj <1 ∗∗ ∗∗

e¯ + e = 1,

e¯

+e

σu

−1

pij∗∗ σu −1

e¯∗∗ =

,

e¯∗i ≤ 1,

(97)

N

e¯∗∗ i ≤ 1, (98)

i=2

e∗ =

,

−1

N i=2

,

−1

j=2,j=i,σj <1

e¯∗ =

,

(αj /αi )σu pij∗∗ σu −1

j=2,j=i,σj >1

= 1+ ∗

−1

j=2,j=i,σj >1

e∗i = 1 + e∗∗ i

(αj /αi )σu pij∗ σu −1

N

e∗i ≤ 1,

(99)

i=2

e∗∗ =

N

e∗∗ i ≤ 1, (100)

i=2

= 1.

(101)

Proof: The limiting shares (91)-(94) follow from (90), (61) and (62). The limiting shares (95)-(100) follow from (90) and (63) and (64). ////

4.3 Homothetic and Non-Homothetic Consumer Preferences The main deﬁciency of many conventional and in fact any homothetic direct utility function is that all the consumer goods have an “income (C)” elasticity of one. Hence, the class of indirect utility functions attracts more attention. A ﬂexible expenditure system satisfying (83) was

36

Bjarne S. Jensen and Mogens E. Larsen

ﬁrst proposed by Leser (1941), and it was related to the indirect addilog by Houthakker (1960). We deﬁne Indirect Addilog Utility as follows [cf. Silberberg & Suen (2001, p. 360), Chung (1994, p. 42)]: αi > 0,

N

αi = 1,

i=2

V (P2 , · · · , PN , C) = γv

ei =

0 < βi < 1, N

αi βi (C/Pi )βi = N αj βj (C/Pj )βj

i=2

N

βi

αi (C/Pi )

αj βj Piβi α β βj j=2 i i Pj

= γv

C βj −βi

N

j=2

β

αj pijj (C/Pi )

βj

,

(102)

−1

j=2

−1 αj βj βj C βj −βi p = α β ij Pi , j=2 i i −1 N ∀j βj = β : ei (ω) = 1 + j=2,j=i [αj /αi ] pij (ω)β

N

(103) (104)

,

ei (ω, L, K) =

N βj −βi C αj βj (ω, L, K) pij (ω)βj 1+ Pi α i βi j=2,j=i

−1

(105) .

The budget system (104), with β = σu −1 > 0, is seen to be a subsystem of the homothetic CES system (90) already characterized in Lemma 3. With βi = βj the non-homothetic indirect addilog utility function has budget shares (105) with terms C/Pi = (1 − s)Y /Pi , cf. (73), where Y /Pi is well-deﬁned in terms of ω, L, K, see (25). With the positivity restrictions above on αi , βi , the budget shares (104-105), always satisﬁes 0 < ei < 1. Without imposing positivity restrictions on all βi , but just assuming N βi < 1, (106) αi > 0, i=2 αi = 1, the Houthakker (addilog) expenditure system (105) is generalized to: βj −βi −1 N ei (ω, L, K) = 1 + j=2,j=i (αj /αi )pij (ω)βj PCi (ω, L, K) .

(107) Still, (106-107) satisﬁes 0 < ei (ω, L, K) < 1. The indirect utility function of this generalized budget system is unknown (and may never be given in closed form). But the empirical application of (107), e.g., Somermeyer (1972), Jensen (1980), works reasonably well; its Engel

General Equilibrium Dynamics of Multi-Sector Growth Models

37

curve patterns allows negative, zero, and positive “income elasticities” (the upper bound may be greater than two), and collectively, the condition (83) is satisﬁed by (107). Finally, we consider the Translog Indirect Utility: N αj > 0, αj = 1, βij = βji , j=2

ln V (P2 , · · · , PN , C) = ln α0 +

N j=2

N

αj ln

N

C C 1 C ln , βij ln + Pi Pj 2 i=2 j=2 Pj (108)

N

αi + j=2 βij ln (C/Pj ) P i Yi , (109) = N N C 1 + k=2 j=2 βkj ln (C/Pj ) N αi + j=2 βij ln(C/Pj ) . ei = N N N αi + j=2 βij ln(C/Pj ) + i=k=2 αk + j=2 βkj ln(C/Pj ) ei =

(110) This translog indirect utility function, Christensen, Jorgenson, and Lau (1971, 1975), Silberberg & Suen (2001, p. 361), Chung (1994, p. 68), is one of the most widely used ﬂexible functional forms in empirical demand analysis. It may be interpreted as a second-order local approximation to an arbitrary indirect utility function. Its budget share is here presented in a proper explicit forms for our purposes. The usual monotonicity property of preferences is satisﬁed by imposing on (108) the condition below (LHS of the implication) [see Chung (1994, p. 69)]: N ∀k : αk + j=2 βkj ln(C/Pj ) > 0 ⇒ ∀i : 0 < ei < 1. (111) This is a suﬃcient condition, cf. (110), for relevant shares, ei , too (RHS of the implication). Rewriting ln (C/Pj ) = ln [C/Pi · Pi /Pj ] = ln (C/Pi · pij ) = ln (C/Pi ) + ln pij , we obtain (110) restated (dividing by its numerator) conveniently as: ⎫ ⎧ N N N ⎪−1 ⎪ ⎪ ⎪ ⎪ βkj ln pij + ln(C/Pi ) βkj ⎪ αk + ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ j=2 j=2 i=k=2 ei (ω, L, K) = 1 + N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α + ln (C/P ) β ⎪ ⎪ i i ij ⎭ ⎩ j=2

.

(112)

38

Bjarne S. Jensen and Mogens E. Larsen

In the non-homothetic case, the terms, C/Pi , (112) are - as mentioned above for the non-homothetic addilog - well-deﬁned in the arguments of ei , (112). By direct inspection of the budget share, (109), a more tractable and simpler suﬃcient condition than (111) for having 0 < ei < 1 is N ∀i : [∀k : Pk ≤ C ∧ βik ≤ (113) j=2 βkj ⇒ 0 < ei < 1] The implication (113) follows here from using ln(C/Pk ) > 0 and the symmetry βij = βji in (109). Accordingly, (113) also applies to budget shares, (112). Requiring in, (113), that all unit prices, Pk , are less than total expenditure, C, is hardly any practical restriction at all; such luxury goods are ignored. For the homothetic translog with the restrictions N (114) j=2 βkj = 0 (k = 2, ..., N ), the budget shares ei become (cf. (112)) N ei (ω) = αi [1 − j=2 βij ln pij (ω) ]−1

(i = 2, ..., N ).

(115)

Besides (114), we must evidently impose further parameter restrictions on βij or indirectly upon the sizes of the relative prices, pij , for having , 0 < ei < 1, in (115). However, we shall not pursue such restrictions, since the homothetic translog has little factual application. With translog, it is the non-homethetic version and its richer Engel curve patterns that makes it attractive/useful. 5 Walrasian General Equilibrium of Multi-Sector Economies The demand side of the multisector economy is expressed by the respective GDP expenditure shares, si , derived from the consumer budget shares, ei , in section 4. The supply side of the economy – operating under constant returns to scale and with full (10-11) and Pareto-eﬃcient factor utilization, (16) – is always summarized by sectorial factor allocation fractions λLi , λKi , which in turn are determined by si and the sectorial cost shares Li , Ki , (33). Theorem 1. The Walrasian equilibrium (competitive general equilibrium) states, – by market clearing prices on the commodity/factor markets and Pareto eﬃcient endowments allocations – are, with homogeneous production functions of degree one, and with any homothetic preferences, given by : ∀k ∈ R+ , ∀ω ∈ Ω, (34), (28), (30), (90), (104), (115), $N N ωδK (ω) si (ω)Ki (ω) si (ω)Li (ω) = Ψ (ω). (116) =ω k= δL (ω) i=1 i=1

General Equilibrium Dynamics of Multi-Sector Growth Models

39

Corollary 1. With CES sector technologies and any homothetic utility function, the Walrasian equilibrium k = Ψ (ω) becomes by (116) and (47): N ω i=1 si (ω)(1 + ci ω 1−σi )−1 = Ψ (ω). (117) k= N 1 − i=1 si (ω)(1 + ci ω 1−σi )−1 Theorem 2. For sector technologies with constant returns to scale and non-homothetic utility functions, Walrasian equilibrium states are given by (34),(28), (30), (105), (107), (112) and N ω si (ω, L, K)Ki (ω) ωδK (ω, L, K) K = Υ (ω, L, K). (118) = Ni=1 = δL (ω, L, K) L i=1 si (ω, L, K)Li (ω) Corollary 2. With CES sector technologies, the Walrasian equilibrium (118) changes with (47) into: N ω i=1 si (ω, L, K)(1 + ci ω 1−σi )−1 = Υ (ω, L, K). (119) K/L = N 1 − i=1 si (ω, L, K)(1 + ci ω 1−σi )−1 Locus expressions (118) and (119) give ω implicitly as a function (graph) of (L,K): ω = Λ(L, K). (120) Proof: The theorems and corollaries are obtained by turning Walras’ law (identity), (34), into the respective Walrasian equilibrium conditions, (116)-(119). Rather than relying on ﬁxed-point methods for searching (iteration) the equilibrium prices (vector) of numerous supply and demand equations of goods and factor markets, our general equilibrium solution procedure is formulated in variables (shares), having simple economic and observable NIPA counterparts. Since relative commodity prices are endogenous variables, they are by construction properly eliminated from our “structural” general equilibrium equations. Hence we end up with a “reduced” form of just one equation (explicit or implicit) between the remaining endogenous relative factor prices of general equilibrium and the exogenous (given) factor endowments. //// The competitive general equilibrium functions, k = Ψ (ω), ω = Λ(L, K), are crucial for inquiring into the statics, comparative statics, and dynamics of multi-sector economies, and they are called the factor endowment-factor price (FEFP) correspondence. Having obtained ω from (116), we can go back through (22), (51), (90) (47), (32), (33) and (24) to get the associated general equilibrium values of all other endogenous variables (sector outputs, allocation fractions of inputs, income shares, relative commodity prices).

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Regarding the shape of the graph of Ψ , (117), it is evident that, if all substitution elasticities are larger than one, σi > 1 for i = 2, ..., N , then the numerator (denominator) expression in (117) will increase (decrease) [cf. Ki , Li (15)], which always ensures that the Walrasian locus Ψ (ω), (117), is monotonically increasing. When all σi are less than one, σi < 1, only a detailed examination will reveal the global and local shape of the graph. Proposition 2. The graphs of the Walrasian equilibrium, Ψ (ω),(117), are in Fig. 1 to be located between the extreme monotonic CES, ωi curves, (46). For any value of the GDP shares, si , and for any size of the sectorial substitution elasticities, σi , i = 1, ..., N, the functions, Ψ (ω), have the limit properties: lim Ψ (ω) = 0,

ω→0

lim Ψ (ω) = ∞,

ω→∞

lim Ψ (ω)/Ψ (ω) = 0,

ω→0

(121)

lim Ψ (ω)/Ψ (ω) = ∞.

ω→∞

(122)

With (117), the elasticities, E(k, ω), of Ψ (ω) have the ﬁnite limits: ∀i : σi < 1 : lim E(k, ω) = lim E(k, ω) = max σi ,

(123)

∀i : σi > 1 : lim E(k, ω) = lim E(k, ω) = min σi ,

(124)

∃i, j : σi > 1 > σj : lim E(k, ω) = lim E(k, ω) = 1.

(125)

ω→∞

ω→0

i

ω→∞

ω→0

ω→0

i

ω→∞

Proof: The limits (121)-(122) are seen immediately from (117). The limits (123)-(125) follow from the formulas (116) and (117): E(k, ω) = Ψ (ω)ω/Ψ (ω) = 1 + E(δK , ω) − E(1 − δK , ω), (126) δ (ω)ω . (127) E(δK, ω) = δK (ω)ω/δK , E(1 − δK , ω) = − K 1 − δK In case of constant si , the numerators of (127), ± δK (ω)ω, go to zero for both ω → 0 and ω → ∞, as is seen by simple calculations. The denominators of (127), δK , 1 − δK , go to s¯ or s for ω → 0 and ω → ∞ in accordance with Table 6. Hence in case of, 0 < s¯ < 1, cf. Table 6 (rows 1,3), the last two terms of (126) go to zero - giving (125). In case of, s¯ = 0, 1, one term of (126) goes to zero, while the other goes to the dominating power, in some cases seen by use of lHospital. This proves (123)-(124). In the case of variable si (ω), we just have to correct the proof above by adding the limits of si (ω)ω/si (ω), which, however, are both zero, since the budget shares ei (ω) have speciﬁc limits ei∗ and ei∗∗ for the respective utility functions, cf. (91)-(100), (104) and (115). ////

General Equilibrium Dynamics of Multi-Sector Growth Models

41

6 Dynamics and Evolution of Multi-Sector Economies In the multi-sectoral planning literature, we often read statements like that of, e.g., Johansen (1974, p. 22): “The growth process is generated by the following factors, all of which are considered to be exogenously determined : a) Total investment, b) Growth in population; working and total, c) Growth in productivity; shifts in production functions over time, d) Changes in exogenous demand; mainly government and net foreign demand.” This broad view of the exogeneity concept refers to exogeneity (separability) assumptions of rather diverse nature. Let us consider those above in reverse order. Some “applied/planning” growth models may include a public sector with government expenditure and taxation assumptions/speciﬁcations, but this sector can properly be excluded for other theoretical/analytical purposes. Similarly, some growth models may include international trade and international factor mobility. Certainly, open and in particular small open economies operate diﬀerently from closed economies, but for obtaining some actual insights about economic evolution, a relevant closed (global) economy growth model may suﬃce. Next, “shifts” in production (or utility) functions refer to “parametric” changes. Since most economic parameters are not “natural constants” and occasionally undergo critical changes, an important object of growth models is to qualitatively understand and identify the crucial parameters involved and their critical numerical values. Regarding labour endowments (population), no attention has so far been directed to its exogenously given size. As a state variable in a dynamic model, it may still be treated, without violating the general equilibrium model above, as evolving exogenously with speciﬁed parameters. On the contrary, however, the time paths of total investments and capital endowments cannot be extrapolated exogenously without violating the general equilibrium solutions of Theorem 1-2. A coherent general equilibrium evolution for the capital endowments in continuous time can only be derived from consistently integrating the endogenous output paths of our sector 1, i.e., the macrodynamic role of the capital good industry (machinery) in growth models of multi-industry-economies must be carefully studied. The equations of factor accumulation for multi-sector growth models, with two primary factors and ﬂexible constant return-to-scale sector technologies, are formally given by (δ : the depreciation rate of capital), dL/dt ≡ L˙ = nL, (128) dK/dt ≡ K˙ = Y1 − δK = Ly1 l1 − δK = L[f1 (k1 )l1 − δk]. (129)

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6.1 General Equilibrium Dynamics with Homothetic Preferences In the general equilibrium models of multi-sector economies, k1 and l1 in (129) are through ω, (16), (116) and (117) uniquely determined by the factor endowments ratio k, cf. (47). Hence, the accumulation equations (128) and (129) become autonomous (time invariant) diﬀerential equations in the state variables L and K and represent a standard homogeneous dynamic system, L˙ = Ln ≡ Lf (k), (130) % & −1 −1 ˙ K = L f1 (k1 [Ψ (k)])l1 [Ψ (k)] − δk ≡ Lg(k), L = 0. (131) As g(k) in (131) is an intricate composite function of k, we rewrite g(k) in alternative forms by (129), (25) and (26): K˙ = s1 Y/P1 −δK = Ls1 (ω + k)f1 (k1 )−δK =

Lk[s1 f1 (k1 )/δK −δ]

= Lg(k),

Lk = K,

(132) (133)

where to succinctly express and decompose the governing functions of capital accumulation (132) - the bounded variable δK (k) is mainly a formal auxiliary term helpful in evaluating concrete cases. From the governing functions g(k) and f (k), (130)–(133), the director function h(k) that controls dk/dt ≡ k˙ takes the form h(k) ≡ g(k) − kf (k),

s1 f1 [k1 [ω(k)]] − (n + δ) , ω(k) = Ψ −1 (k). (134) k˙ = h(k) = k δK [ω(k)] The dynamic system (134) in k is diﬃcult to evaluate quantitatively and generally analytically intractable; e.g., if σi = σj , then, k = Ψ (ω), (116) can not be inverted (although Ψ −1 exists) in closed form. But k = Ψ (ω), (116), are continuously diﬀerentiable functions of ω, and dynamics in k can be converted into dual autonomous dynamics in ω: ω˙ =

h(Ψ [ω]) h(k) k˙ ≡ (ω). = = Ψ (ω) dk/dω dk/dω

(135)

Hence we get (cf. (134))

Ψ (ω) s1 f1 [k1 (ω)] − (n + δ) (ω) = δK (ω) Ψ (ω)

s1 f1 [k1 (ω)] ω − (n + δ) . = δK (ω) E(k, ω)

(136) (137)

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43

With CES technologies, from (137), (32), (30), (47), (81), we have ⎤ ⎡ σ1 σ1 −1 1−σ1 1/(σ1 −1) (1 + c1 ω ) Ψ (ω) ⎣ s1 γ1 a1 − (n + δ)⎦ (138) (ω) = N 1−σi −1 Ψ (ω) ) i=1 si (ω)(1 + ci ω with Ψ (ω) given in (117).

6.2 Existence and Uniqueness of Steady States or Persistent Growth The complete set (family) of k(t) solutions to the dynamic systems (134) is qualitatively described and classiﬁed by the following theorem. Theorem 3. The multi-sector growth models (128) have no positive, stationary k(t)-solution [k(t) = 0 is attractor], iﬀ ¯ 1 < ([n + δ] /s1 )δK (k) ∀k > 0 : b

(139)

and have at least one steady state [ray path in (L, K)-space], iﬀ [cf. (3)] ∃k > 0 : ([n + δ]/s1 )δK (k) ∈ J1 .

(140)

The stationary capital-labor ratios k(t) = κ for all t are obtained by f1 [k1 (κ)] = ([n + δ]/s1 )δK (κ).

(141)

With existence (140), a suﬃcient condition for a unique root of h(k) is ∀k > 0 : E(h(k)/k, k) < 0 ⇔ ∀ω ∈ Ω : E(k, ω) ≥ 1.

(142)

The time paths of the growth model solutions, k(t), display persistent growth – limt→∞ k(t) = ∞ – if and only if ∀k > 0 : b1 > ([n + δ] /s1 ) δK (k).

(143)

Proof: The set of solutions to (134) depends entirely on the shape of the director function, h(k), and the number of roots of h(k). The existence of nonzero roots requires that ([n + δ]/s1 )δK (k) belongs to the range of f1 as stated in (140). If no positive root exists, we have either the case (139) with origo as attractor, or the case (143) with persistent growth. If a root of (141) exists, a unique attractor in the interval stated in (140) always occurs with a global negative sign of the elasticity, E(h(k)/k, k) < 0, which can be derived from (134); cf. (142), (123)(125) and Jensen [1994, p. 138, p. 129], [2003, p. 73]. ////

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6.3 Dynamics of MSG Models with CES Sector Technologies Qualitative properties of the family of Walrasian general equilibrium solutions k(t) in multi-sector growth models with CES technologies are summarized as follows: Theorem 4. For the multi-sector growth models (128)-(131) with CES technologies, the suﬃcient conditions for the existence of at least one positive steady-state solution are [no positive, attractive, steady state solution κ (141) exists with the inequalities of (144-145), RHS, reversed ] : σ1

¯ 1 = γ1 a σ1 −1 > (n + δ)/s1 . ∀i σi < 1 : b 1 With only assuming σ1 < 1, (144) is generalized to, [β = 1 + σ1 (n + δ)s (n + δ)β for σu < 1, σ1 −1 ¯ = > b1 = γ1 a1 for σu > 1. n+δ s1

(144) 1−s1 ∗ s1 e ]

(145)

If σ1 ≤ 1 (suﬃcient condition), persistent growth of k(t) is impossible. If σ1 > 1, necessary and suﬃcient conditions for limt→∞ k(t) = ∞ are: σ1

∀i σi > 1 : b1 = γ1 a1σ1 −1 > (n + δ)/s1 . With only assuming σ1 > 1, (146) is generalized to, [β¯ = 1 + σ1 (n + δ)¯ s (n + δ)β¯ for σu > 1, = b1 = γ1 a1σ1 −1 > n+δ for σu < 1, s1

(146) 1−s1 ∗ ¯] s1 e

(147)

except that (147) is occasionally not suﬃcient for small initial values. Proof: The proof proceeds with the dual version, (ω). The term Ψ (ω)/Ψ (ω) has no inﬂuence on these limit analyses, cf. Proposition 2. Assume ﬁrst that σ1 < 1. The fraction in the large bracket (138) goes to zero for ω → ∞; hence, there are no permanent increasing solutions of ω(t). If (144)-(145) are satisﬁed, then this fraction passes at least once monotonically through the constant, when ω goes from zero to inﬁnity. The diﬀerence s in the constant comes from the denominator taking values 1 or s for ω → 0, depending on the size of other σi . If the inequalities are reversed, then (ω) is negative. The role of s in (145) follows from (138), (31) and (91)-(101). Second, assume σ1 > 1: The fraction in the large bracket (138) goes to inﬁnity for ω → 0. If and only if (146)-(147) are satisﬁed, then this fraction eventually remains above the constant, when ω goes from zero to inﬁnity. The diﬀerence s1 in the constant comes from the denominator taking values 1 or s¯ for ω → ∞, depending on the size of other σi . Hence (ω) is positive for large values of k. If the inequalities are

General Equilibrium Dynamics of Multi-Sector Growth Models

45

reversed, then (ω) eventually becomes negative. The necessary conditions (146) are also suﬃcient, as k(k) is, with respectively σi > 1 and σ1 > 1, monotonically decreasing, but remain above RHS values, (146). The role of s¯ in (147) follows from (31), (138) and (91-101). //// Theorem 4 shows clearly that the global existence problem of steady states (κ) or persistent growth of k(t) depends on the size of the key parameters: σi , a1 , γ1 , s1 , n, δ, s¯. The accumulation parameters (s1 ,¯ s, n, δ) play some roles; but the fundamental role of the technology parameters in the capital good industry (σ1 , γ1 , a1 ) - altering the long-run k(t) solutions of multi-sectoral general equilibrium growth models complies with observation and intuition. Evidently, the strategic importance ascribed to the capital good industries by economic historians and the general public, makes good economic sense, at least for the closed (global) economy. The single most important parameter in Theorem 4 is the substitution elasticity in the capital good sector, σ1 . It must clearly be larger than one for persistent growth to take place. However, the “total productivity” parameter γ1 in the capital good sector aﬀects all the stated conditions (144-147), and they can all be violated by giving γ1 any value between 0 and ∞. A larger TFP parameter of the capital good sector γ1 may give a “big push”, cf. Murphy et al. [1989], Parente & Prescott [1999], Prescott [1998]. But if we restrict γ1 = 1 and if σ1 2, then (146) will usually be satisﬁed with the other parameters, in particular, with high saving rates. The key role of the technology in the capital good industry had escaped the “mainstream” literature on two- and multi-sector growth models, cf. Jensen (2003, p.75). As to empirical evidence, the theoretical general equilibrium predictions of Theorem 4 tally with studies of long-run growth conducted by, Rosenberg (1963), De Long & Summers, (1991), Rebolo (1991), and Jones [1994]. In particular, high rates of equipment investment (“mechanization”) are prime determinants for national growth performance (per capita growth). Furthermore, the making of various equipments become eventually highly mechanized by making various engines (steam, combustion, electric) “cheap as well as good,” cf. Mokyr [1990, p. 87]. This supports factor substitution and mechanization subsequently in the consumer good industries. In this way, the capital good (multi-purpose machinery/equipment) is a “Lever of Riches” (productivity and per capita growth) in several sectors with the capital good industry itself and its technology parameters being naturally of primary importance for sustaining the economic growth process - as here mathematically demonstrated in a Walrasian general equilibrium framework.

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On advances in technology and economic evolution, Usher (1954, p. 10) writes: “It is important not to presume a continuous development of technology at a constant rate, but it is important, also, to recognize that the process of social evolution consists in part in the cumulative development of science and technology. We need both a general understanding of the process or processes, and, when records make it possible, a documented account of the history of particular periods and particular achievements” and (p. 380) “The technique of interchangeable -part manufacture was thus established in general outline before the invention of the sewing machine or harvesting machinery. The new technique was a fundamental condition of the great achievements realized by inventors and manufacturers in those ﬁelds. It made it possible to place the sewing machine in the home and it generalized the use of harvesting machinery of McCormick and Deere with astonishing rapidity. American engineering and manufacturing ﬁrms took the lead in this general development, achieving distinctive results over an important ﬁeld that was steadily enlarged decade after decade. The group of machine tools became more and more automatic, and it became possible to build highly specialized machinery for manufacturing ﬁrms. Great reﬁnements of execution were achieved with the simplest labor of attendance. These highly developed machine tools are the most distinguished “iron men” of the modern industrial world, for they make possible that substitution of machinery for labor that is so happily described as eﬀecting a “transfer of skill”.” Can historical stages (parametric changes) of increasing substitution elasticities in the industries of consumer or especially capital goods be better and more eloquently described? Multi-sectoral dynamics oﬀer the same economic message about key parameters behind the patterns of industrial evolution. 6.4 General Equilibrium Dynamics with Non-Homothetic Preferences The qualitative insight gained by Theorem 4 and the discussion above about the dynamic role of critical parameter values is not conﬁned to economies of homothetic consumer preferences. Evidently, with nonhomothetic preferences, the factor accumulation equations (128), 129), 132) and (133) still apply,

s1 f1 (k1 [ω(Λ[L, K])]) ˙ ˙ L = nL, K = Y1 − δK = K −δ . (148) δK (L, K)

General Equilibrium Dynamics of Multi-Sector Growth Models

47

Although the equations (148), cannot be reduced to a single equation in the capital-labor ratio, k, and certainly neither in the wage-rental ratio ω, (135-138), the accumulation equations, (148), still represent a well-deﬁned dynamic system in the state variable, L, K, by which, without explicit solutions, the logarithmic time derivative of k(t) is easily obtained, cf. (134): ˙ −L = kˆ = d ln k = k = s1 f1 (k1 [ω(Λ[L, K])]) − (n + δ). K δK (L, K) k dt

(149)

Thus the qualitative question of steady state or persistent growth of k(t) from (148) similarly depends, cf. (143), on the condition b1 > ([n + δ] /s1 ) δK (L, K) for all L and K.

(150)

The conditions of (147) still applies, but here with s¯ dependent of (L, K), cf. (105), (107), (112). Although there are obviously quantitatively great diﬀerences for economies with non-homothetic preferences (income elasticities diﬀerent from one, Engels law), it is seen from (149)-(150) that the critical role of the capital good industry and its substitution elasticity carry over from Theorem 4. 7 Persistent Growth and Asymptotic Growth Rates To complement the persistent growth solutions of the state variable k(t) or ω(t) with disaggregate information about the general equilibrium evolution for sectorial and other endogenous per capita variables, we characterize the respective time paths by their asymptotic growth rates [ˆ ω (t) ≡ ω/ω(t), ˙ etc.]: Theorem 5. With (146)-(147), the long-run growth rates of k(t) and ω(t) in Walrasian multi-sector growth models (128) with CES technologies are: ∀i σi > 1 : lim ω ˆ= t→∞

s1 b1 − (n + δ) ; lim kˆ = s1 b1 − (n + δ). (151) t→∞ min {σi }

1 ∗ ¯] With only assuming σ1 > 1, (151) is generalized to, [β¯ = 1 + 1−s s1 e s1 b1 b1 /β¯ − (n + δ) for σu > 1, − (n + δ) = lim ω ˆ = lim kˆ = for σu < 1. b1 − (n + δ) t→∞ t→∞ s¯ (152)

With (151), the long-run sectorial and per capita growth rates are i ) = lim (y/P i ), lim kˆi = lim yˆi = lim (w/P

t→∞

t→∞

t→∞

t→∞

(153)

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Bjarne S. Jensen and Mogens E. Larsen

where ∀i : σi > 1 : lim kˆi = t→∞

σi [s1 b1 − (n + δ)]. min {σj }

(154)

With (152), the long-run sectorial and per capita growth rates are σi b1 /β¯ − (n + δ) for σu > 1, ˆ ˆ (155) lim ki = σi lim k = for σu < 1. σi [b1 − (n + δ)] t→∞ t→∞ If only the capital good sector has a substitution elasticity, σ1 > 1, then kˆ → b1 −(n+δ), kˆ1 → σ1 [b1 −(n+δ)], kˆi → σi [b1 −(n+δ)], (156) 1 ) = lim (y/P lim kˆ1 = lim yˆ1 = lim (w/P (157) 1 ), t→∞

t→∞

t→∞

t→∞

whereas the output of all other sectors will ultimately stagnate, σi

lim yˆi = 0 and lim yi = γi (1 − ai ) σi −1 for i = 1.

t→∞

t→∞

(158)

Proof: Theorem 5 follows from Theorem 3, (146), combined with (135), (137) and (124), and using (45)-(47), (24) and (16). Thus, by (137):

ˆ = [ lim E(k, ω)]−1 [s1 b1 /s − (n + δ)] lim ω

t→∞

ω→∞

(159)

The FEFP correspondence k = Ψ (ω) next gives kˆ = E(k, ω)ˆ ω;

kˆi = σi ω ˆ,

yˆi = ki kˆi

(160)

which holds generally with CES. These relations and limits establish the relevant asymptotic growth rates in Theorem 5. //// As Theorem 5 supplements Theorem 3 and Proposition 2, only a few ˆ (151) and (152) remarks are needed. The asymptotic growth rates of k, correspond, respectively, to those implied by (146) and (147). Evidently, with more industries to be highly mechanized (σi > 1) and hence larger ˆ The same applies s¯, the slower will be the overall accumulation rate, k. to the sectorial kˆi , (154)-(155); but for kˆi , what also matters is the elasticity of Ψ and its own σi , cf. (160). If maximum growth of per capita consumption is the goal, then the ranking with, all σj > σ1 > 1, is preferred – which contributes to mechanizing and maintaining the growth rate of all consumer goods and thereby increases the welfare per capita in any numeraire (sectorial) good, (153). The other extreme is capital accumulation for its own sake (156)-(158); capital accumulation, causing increasing wage-rental rates, makes it impossible to avoid increasing sectorial (ki ) anywhere, even though diminishing returns (with σi < 1) eventually terminate increases in sectorial labor productivity. Thus, even with homothetic preferences and only price elasticities involved on the demand side, diverse patterns of industrial growth may emerge with diﬀerent CES technologies on the supply side of a multisector economy.

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8 Final Comments The opinion of Ricardo (1965, p.263-69) on capital goods (machinery) was the following: “Ever since I ﬁrst turned my attention to questions of political economy, I have been of opinion that such an application of machinery to any branch of production as should have the eﬀect of saving labour was a general good, accompanied only with that portion of inconvenience which in most cases attends the removal of capital and labour from one employment to another. -These were my opinions, and they continue unaltered, as far as regards the landlord and the capitalist; but I am convinced that the substitution of machinery for human labour is often very injurious to the interests of the class of laborers. - The statements which I have made will not, I hope, lead to the inference that machinery should not be encouraged. To elucidate the principle, I have been supposing that improved machinery is suddenly discovered and extensively used; but the truth is that these discoveries are gradual, and rather operate in determining the employment of the capital which is saved and accumulated than in diverting capital from its actual employment”. His contemporary von Th¨ unen [1850 (1930, p.499)] had similar opinion: “W¨ahrend man in Europa den gedr¨ uckten Zustand der arbeitenden Klasse so ha¨ uﬁg der zunehmenden Anwendung von Maschinen zuschreibt, wird in dem gesellschaftlichen Zustand, den wir hier vor Augen haben, die Lage der Arbeiter immer bl¨ uhender and gl¨ anzender, je ausgedehnter beim Anwachsen des Kapitals die Anwendung von Maschinen wird”. Both issues - factor reallocations and capital accumulation combined with the GDP growth per capita of multi-sector economies - are still with us and will continue to be so, in Europe and globally. In such historical and future human circumstances, it may help (as in natural sciences) - our spirit, knowledge, daily problems and nerves – being able to basically understand the logic and to formally describe the economic laws of motion (change). Mathematical models of general equilibrium dynamics for growing economies serve such purposes, as attempted in this paper. A still widely held didactic view is the following [Myrdal (1973, p. 182)]: “The emergence and triumph, about a hundred years ago, of the theory of marginal utility and its embodiment in the static equilibrium conception marked an end to the interest the classical

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authors, and, of course, Marx, had shown in the long-term problems of growth and development. The focus was put on problems about static eﬃciency and static allocation of resources, relative to given consumer preferences as expressed by aggregate demand.” As quotations above show, we do not disparage insights of classical economic writings; but some of their modes of analysis have little in common with neoclassical or modern equilibrium methodology. The signiﬁcance of any equilibrium (“static”) analysis is that it provides a solution principle for the economic variables that are involved with the interrelated commodity and factor markets. Moreover, “static efﬁciency” in allocation is not a rather trivial matter, economically or mathematically. In fact, allocative (Pareto) eﬃciency implemented by a common marginal rate of substitution (MRS), ω = w/r, gave us the basic economic variable to reduce the dimension of the solution space for multi-sector economies. The FPCP correspondence, pij (ω), consistently linked the interrelated goods and factor markets. Total factor endowments and the parameters of production and utility functions ultimately determined the set (locus) of Walrasian (general) equilibria. In general equilibrium dynamics, we similarly solve for equilibrium time paths. As was seen in section 6, the dynamic system could certainly never be solved without ﬁrst having obtained the Walrasian equilibrium solutions, k = Ψ (ω), or, ω = Λ(L, K). In economics, as in physics (mechanics), there was never any progress in dynamics before the basic problems of static were resolved. No economic growth models or any “magniﬁcent dynamics” (in mathematical sense) were ever analyzed in the classical period. We have come a long way and hopefully reached a higher vantage point, which oﬀer a better view and overall understanding of the roads passed. In closing, let us observe that the dynamics of growing economies is now a worldwide ﬁeld of study; extensions of this paper on both the demand/consumer and supply/producer side of MSG models should be straightforward.

General Equilibrium Dynamics of Multi-Sector Growth Models

51

Appendix The comparative static analysis of exogenous factor endowment, (L, K), variations for Walrasian equilibria with CES sector technologies is helpful for the economic understanding of the sectorial allocation implications of critical parameter values. As benchmarks, the asymptotic factor allocations of multi–sector general equilibrium economies with various GDP income shares, δK and sectorial factor allocations, λKi , λLi , are calculated as follows: Lemma A1. For the CES multi-sector competitive general equilibrium economy, limits of factor allocation fractions and factor income shares - for a demand side with constant ei , (σu = 1), (86), (89) - are: Table 6 k 1 > σi = σmax 1 > σi = σmax 1 < σi = σmin 1 < σi = σmin

li → 0 li → 1 li → ss¯i li → si

→ 0

λKi → ssi λKi → si λKi → 0 λKi → 1

k δK → s δK → 1 δK → s δK → 0

li → ssi li → si li → 0 li → 1

→∞

λKi → 0 λKi → 1 λKi → ss¯i λKi → si

δK → s¯ δK → 0 δK → s¯ δK → 1

where s and s¯ are given by (31). If all the technologies have the same σi , then the limits above become: Table 7 k σi = σ < 1 σi = σ > 1

→ 0

→ ∞

k

li → l¯i λKi → si δK → 1 li → si λKi → λKi δK → 0 li → si λKi → λKi δK → 0 li → l¯i λKi → si δK → 1

where li ≡ si ci /

sj cj and λKi ≡ (si /ci )/

sj /cj .

Proof: From Lemma 1, (31), (32), (47), we get, si si + δK = 1−σ i 1 + ci ω 1−σi 1 + ci ω σ >1 σ <1 i

i

For ω → ∞, the ﬁrst sum goes to 0, and the second goes to s. For ω → 0, the second sum goes to zero, and the ﬁrst goes to s. From (47), we get Ki → 0 for ω → ∞ and σi < 1, and for ω → 0 and σi > 1. Furthermore, we get Ki → 1 for ω → ∞ and σi > 1, and for

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Bjarne S. Jensen and Mogens E. Larsen

ω → 0 and σi < 1. Together with (39), we get the limits for λKi , and together with (38), we get the limits for li . //// Lemma A2. In the case of the CES, (88), and the indirect Addilog utility functions, (103), with the GDP shares, si (ω), (29), depending on ω, the same conclusions (Tables 6 and 7) hold with limits, e¯∗i , e∗∗ i , e, e, given by (97)-(101) that eﬀects s and s¯ from (31).

Acknowledgements For valuable comments on earlier drafts, we are indebted to two anonymous referees, to participants at the: DEGIT VIII conference, Helsinki, May 2003; Seminars: WiSo,University of Rostock, October 2003; WIF/ ETH, Z¨ urich, December 2003; School of Mathematical Sciences, Peking University, December 2003; School of Management and Engineering, Nanjing University, December 2003; SIPEB, Aoyama Gakuin University, Tokyo, March 2004; RIEB, Kobe University, March 2004; IEAS, Academia Sinica, Taiwan, March 2004; Dept.of Econ., National University of Singapore, March 2004; School of Economics and Management, Lund University, April 2004.

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David, P. A., and Scadding, J. L. (1974): “Private Savings: Ultrarationality, Aggregation, and ’Denison’s Law’.” Journal of Political Economy 82: 225-249. Deaton, A., and Muelbauer, J. (1994): Economics and Consumer Behavior. Cambridge: Cambridge University Press. Debreu, G. (1959): Theory of Value. New York: J. Wiley. De Long, J. B., and Summers, L. H. (1991): “Equipment Investment and Economic Growth.” Quart. J. Economics 106: 445-502. Denison, E. F. (1958): “A Note on Private Saving.” Review of Economics and Statistics 40: 261-267. Dhrymes, P. J. (1962): “A Multisectoral Model of Growth.” Quarterly Journal of Economics 76: 264-278. Echevarria, C. (1997): “Changes in Sectoral Composition Associated with Economic Growth.” International Economic Review 38: 431-452. Frisch, R. (1959): “A Complete Scheme for Computing All Direct and Cross Demand Elasticities in a Model with Many Sectors.” Econometrica 27: 177-196. Frisch, R. (1965): Theory of Production. Dordrecht, Holland: O. Reidel Publishing Company Gravelle, H., and Rees, R. (2004): Microeconomics 3. ed. Prentice Hall. Harberger, A. C. (1962): “The Incidence of the Corporation Income Tax.” Journal of Political Economy 70: 215-240. Houthakker, H. S. (1957): “An International Comparison of Household Expenditure Patterns, Commemorating the Centenary of Engel’s law.” Econometrica 25: 532-551. Houthakker, H. S. (1960): “Additive Preferences.” Econometrica 28: 244-257. Houthakker, H. S. (1965): “A Note on Self-Dual Preferences.” Econometrica 33: 797-801. Inada, K. (1963): “On a Multi-Sector Model of Economic Growth: Comments and a Generalization.” Review of Economic Studies 30: 119-27. Islam, N. (1995): “Growth Empirics: A Panel Data Approach.” Quarterly Journal of Economics 110: 1127-1170. Jensen, B.S. (2003): “Walrasian General Equilibrium Allocations and Dynamics in Two-Sector Growth Models.” German Economic Review 4: 53-87. Jensen, B.S., Richter, M., Wang, C., and Alsholm, P.K. (2001): “Saving Rates, Trade, Technology, and Stochastic Dynamics.” Review of Development Economics 5: 182-204. Jensen, B.S., and Wang, C. (1999): “Basic Stochastic Dynamic Systems of Growth and Trade.” Review of International Economics 7: 378-402. Jensen, B.S. (1994a): The Dynamic Systems of Basic Economic Growth Models. Dordrecht: Kluwer Academic Publishers. Jensen, B.S., and Larsen, M.E. (1994b): “Growth and Long-Run Stability.” Acta Applications of Mathematics 9: 219-37; or in Jensen (1994a).

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Jensen, B.S. (1980): “Expenditure System for the Danish Household Sector.” Symposium i Anvendt Statistik (1980). Lyngby, Denmark: NEUCC: 263-296. Johansen, L. (1974): A Multi-Sectorial Study of Economic Growth. 2. Ed. Amsterdam: North-Holland. Jones, R.W. (1965): “The Structure of Simple General Equilibrium Models.” Journal of Political Economy 73: 557-72. Jorgenson, D. W. (1961): “The Structure of Multi-Sector Dynamic Models.” International Economic Review 2: 276-293. Judd, K. L. (1998): Numerical Methods in Economics. Cambridge (Mass.): MIT Press, Mass. Kongsamut, P., Rebelo, S., and Xie, D. (2001): “Beyond Balanced Growth.” Review of Economic Studies 68: 869-882. Koopmans, T.C. (1960): “Stationary Ordinal Utility and Impatience.” Econometrica 28: 287-309. Laitner, J. (2000): “Structural Change and Economic Growth.” Review of Economic Studies 67: 545-561. Leser, C.E.V. (1941): “Family Budget Data and Price Elasticities of Demand.” Review of Economic Studies 9: 40-57. Mas-Colell, A., Whinston, M.D., and Green, J.R. (1995): Microeconomic Theory. New York: Oxford University Press. Meckl, J. (2002): “Structural Change and Generalized Balanced Growth.” Journal of Economics 77: 241-266. Mokyr, J. (1990): The Lever of Riches: Technological Creativity and Economic Progress. New York: Oxford University Press. Murphy, K. M., Shleifer, A., and Vishny, R. (1989): “Industrialization and the Big Push.” Quarterly Journal of Economics 97: 1003-1026. Myrdal, G. (1973): Against the Stream. Critical Essays on Economics. New York: Pantheon Books. Pais, A. (1986): Inward Bound. Of Matter and Forces in the Physical World. Oxford: Oxford University Press. Parente, S.L., and Prescott, E.C. (1994): “Barriers to Technology Adoption and Development.” The Journal of Political Economy 102: 298-321. Pasinetti, L. L. (1981): Structural change and Economic Growth. Cambridge: Cambridge University Press. Prescott, E. C. (1998): “A Theory of Total Factor Productivity.” International Economic Review 39: 525-551. Rebelo, S. (1991): “Long-Run Policy Analysis and Long-Run Growth.” Journal of Political Economy 99: 500-21. Ricardo, D. (1965): The Principles of Political Economy and Taxation. London: J. M. Dent & Sons. Rosenberg, N. (1963): “Technological Change in the Machine Tool Industry, 1840-1910.” The Journal of Economic History 23: 414-443. Samuelson, P. A. (1953): “Prices of Factors and Goods in General Equilibrium.” Review of Economic Studies 21: 1-20.

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Samuelson, P. A. (1965): “Using Full Duality to Show that Simultaneously Additive Direct and Indirect Utilities Implies Unitary Price Elasticity of Demand.” Econometrica 33: 781-796. Scarf, H. E. (1967): “The Approximation of Fixed Points of a Continuous Mapping.” SIAM Journal of Applied Mathematics 15: 1328-43. Scarf, H. E., and Hansen, T. (1973): The Computation of Economic Equilibria. Yale: Yale University Press. Shoven, J.B., and Whalley, J. (1984): “Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey.” Journal of Economic Literature 22: 1007-1051. Shoven, J.B., and Whalley, J. (1992): Applying General Equilibrium. Cambridge: Cambridge University Press. Silberberg, E., and Suen, W. (2001): The Structure of Economics, A Mathematical Analysis. Singapore: McGraw-Hill. Somermeyer, W.H., and Langhout, A. (1972): “Shapes of Engel Curves and Demand Curves: Implications of the Expenditure Allocation Model, Applied to Dutch Data.” European Economic Review 3: 351-386. Stiglitz, J.E., and Uzawa, H. (eds.) (1969): Readings in the Modern Theory of Economic Growth. Cambridge (Mass.): M.I.T. Press. Stolper, W. F., and Samuelson, P. A. (1941): “Protection and Real Wages.” Review of Economic Studies 9: 58-73. Tinbergen, J. (1960): “Optimum Savings and Utiility Maximation Over Time.” Econometrica 28: 481-489. Tobin, J. (1980): Asset Accumulation and Economic Activity. Yrj¨ o Jahnsson Lectures. Oxford: Basil Blackwell Publishers. Usher, A. P. (1954): A History of Mechanical Inventions. Revised Edition. New York: Dover Publications. Uzawa, H. (1969): “Time Preference and the Penrose Eﬀect in a Two-Class Model of Economic Growth.” Journal of Political Economy 77: 628-652. Uzawa, H. (1968): “Time Preference, the Consumption Function, and Optimum Asset Holdings.” in Papers in Honour of Sir John Hicks: Value, Capital, and Growth, edited by J.N. Wolfe, pp. 485-504. Edinburgh: University Press. Uzawa, H. (1973): “Towards a Keynesian Model of Monetary Growth.” p. 53-74, in: Mirrlees, J. A. & Stern, N. H.: Models of Economic Growth. London: Macmillan. Varian, H. R. (1992): Microeconomic Analysis., 3. ed. New York: W. W. Norton. von Th¨ unen, J. H. (1960): Der isolierte Staat in Beziehung auf Landwirtschaft und National¨ okonomie. Hamburg, (1826); 2.ed., Band 1: Rostock, 1842, Band 2: Rostock, 1850. Der isolierte Staat (Band 1-2): Sammlung sozialwissenschaftlicher Meister. Waentig, H., ed. Jena (1930): Verlag Gustav Fischer, Jena. Translations: Wartenberg, C. M.: Von Th¨ unen’s Isolated State, Oxford, 1966. Dempsey, B. W.: The Frontier Wage, Jesuit Studies, Chicago, 1960.

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Walras, L. (1954): Elements of Pure Economics. Jaﬀ, W. (trans.) (1954): London: George Allen and Unwin. Wan, H.Y., Jr. (1971): Economic Growth. New York: Harcourt Brace Jovanowich. Wold, H. (1953): Demand Analysis. New York: J. Wiley and Sons. Addresses of authors: – Bjarne S. Jensen, Department of Economics, Copenhagen Business School, Solbjerg Plads 3, 2000 Frederiksberg, Denmark (email: [email protected]); – Mogens E. Larsen, Department of Mathematics, Institute for Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark (e-mail: [email protected])

J. Econ. (2005) Suppl. 10: 57-76

Common Markets, Economic Growth, and Creative Destruction Tapio Palokangas Received April 15, 2003; Revised version received August 1, 2004 c Springer-Verlag 2005

Economic integration is examined in a multi-economy Schumpeterian growth model where economies diﬀer in their research environment, and consequently in the productivity of R&D. It is shown that economies with more or less the same productivity of R&D integrate. In equilibrium, there can be many common markets with diﬀerent growth rates as well as stagnating economies with decreasing relative income. A small economy with low incentives to save can avoid stagnation, if its R&D is so productive that a common market with a positive growth rate can accept it as a member. Keywords: endogenous growth, convergence, economic integration JEL classiﬁcation: F15, F21, O40.

1 Introduction It is known that diﬀerent institutions are reﬂected in the diﬀerent levels of productivity in the R&D sector. Many historically determined factors, e.g., bad working practices, a low standard of legal institutions and a high rate of criminality, decrease the productivity of R&D.1 Because institutions change slowly, country-speciﬁc diﬀerences in the productivity of R&D may emerge for a long time. In the EU, new members will have weaker research environment than the old members. It is instructive to examine the consequences of this change. In this study, we show that the expansion of a common market to economies with lower productivity in R&D has two opposing eﬀects on welfare. New members increase welfare through a wider variety of products, but on the other hand they decrease the average productivity in R&D in the common market, which lowers R&D, the growth rate 1

See e.g., Barro (1998).

58

Tapio Palokangas

and welfare. When these two eﬀects are balanced, an optimal common market obtains. In this study, we also examine how country-speciﬁc diﬀerences in the productivity of R&D aﬀect growth and economic integration. We give one possible explanation for the following stylized facts: 1. There has been a wide dispersion in cross-country growth rates;2 2. many countries have had a common long-term growth rate;3 3. some small countries have been able to grow much faster than others; and4 4. some countries started to grow much faster after establishing a common market with a greater economy.5 Common market studies have mainly concerned two aspects of integration: (i) free trade, and (ii) the equalization of the rates of return. Walz (1998) shows that the integration of a third, technologically lagging economy into a common market of two technologically advanced economies causes a reallocation of resources which stimulates overall growth. According to Baldwin and Forslid (2000), trade liberalization stimulates growth via increased competition in the R&D and ﬁnancial sectors. Peretto (2003) shows that integration slows down growth via higher ﬁxed costs for a greater variety of goods, but speeds up growth via greater economies of scale. He argues that the latter eﬀect tends to dominate and growth and welfare will rise. In contrast to these papers, we focus on a third aspect of economic integration, the reallocation of research activity among the member economies. From that viewpoint, common markets generate interesting development patterns. For instance, an economy, which did not undertake R&D before, may start doing so if it joins a common market. Howitt (2000) composes a model of creative destruction with international technology spillover. He speciﬁes institutional aspects that impinge directly on the incentive to invest and innovate as shadow subsidies (or taxes) and obtains the following results. Economies with R&D converge to parallel growth paths with diﬀerent productivity levels, but the other economies stagnate. The world growth rate depends positively on the R&D subsidy rate in all economies. In Howitt (2000), the main reason for the convergence of the national growth rates is that every economy with R&D gradually adopts the same frontier technology. In this paper, we show that the growth rates of all economies belonging to the same common market converge, although the economy-speciﬁc lev2 3 4 5

See e.g., Barro and Sala-i-Martin (1995). See e.g., Howitt (2000). E.g., Ventura (1997) and the East Asian Miracle. E.g., Ireland and Portugal after joining the EU.

Common Markets, Economic Growth, and Creative Destruction

59

els of productivity in R&D were exogenously given. Hence, no transfer of technology is needed to explain growth convergence. The remainder of this paper is organized as follows. Section 2 constructs the basic model, for which section 3 presents the equilibrium properties. Section 4 examines households that save by investing in R&D ﬁrms in the same common market. Section 5 analyzes the eﬀects of economic integration.

2 The Model 2.1 Technology The borders of common markets impose implicit costs on both trade and investment ﬂows. We take this property to the extreme, for simplicity, and assume that such costs are at prohibitive levels. Since there is no trade or investment between common markets, the model has the nice property that the price levels of diﬀerent common markets are totally independent. In the model, a great number J of economies exists. Any set of these economies, Γ ⊂ {1, ..., J}, can form a common market. In each economy j, there is one ﬁrm at a time which produces intermediate good j from local labor. The R&D ﬁrms located in economy j ∈ Γ attempt to improve intermediate good j. They employ labor everywhere in the same common market Γ and ﬁnance their expenditure wholly by issuing shares. We assume that the level of productivity λj in R&D is exogenously given and diﬀers throughout the economies j = 1, .., J. When λj is high (low), we say that research environment is good (bad) in economy j. We assume that research ﬁrms in economy j try to make a better version of intermediate good j and employ labor everywhere in the common market to which economy j belongs. The arrival rate of innovations in the R&D process is therefore given by Λj = λk zjk dk, (1) k∈Γ

where zjk is the demand for labor in economy k by R&D ﬁrms located in economy j, and λk the (exogenously given) productivity of R&D in economy k. We denote time by τ and characterize technological change in economy j by a Poisson process qj as follows. During a short time interval dτ , there is an innovation dqj = 1 with probability Λj dτ , and no innovation dqj = 0 with probability 1 − Λj dτ . In the advent of success, a R&D ﬁrm drives the old local monopoly out of the market and takes over the production of the intermediate good. R&D ﬁrms ﬁnance their

60

Tapio Palokangas

investment by issuing shares. Each new generation of any intermediate good provides exactly ε > 1 times as many services as the product of the generation before it. We denote the serial number of technology in economy k by tk . The level of productivity in the production of intermediate good k, Bk (tk ), is determined by the currently most advanced technology tk . The invention of a new technology raises tk by one and the level of productivity Bk (tk ) by ε > 1. This implies Bk (tk ) = Bk (0)εtk .

(2)

2.2 Consumption-Good Producers The size of common market Γ is measured by . θ(Γ ) = dk ≤ J.

(3)

k∈Γ

A large number of competitive ﬁrms produces the consumption good from all intermediate goods k made in the same common market Γ through Cobb-Douglas technology with constant returns to scale, ' C = µ(θ(Γ )) (Bk xk )1/θ(Γ ) , µ > 0, (4) k∈Γ

where C is consumption in common market Γ , xk the demand for intermediate input k, Bk the productivity of input k and θ(Γ ) the size of Γ . The function µ(θ) with µ > 0 characterizes the property that a wider variety of products (i.e., a bigger θ) yields more services to the households and thereby increases welfare.6 This is the main motivation of economic integration. The representative consumption-good ﬁrm in common market Γ maximizes its proﬁt . Π = PC − pk xk dk k∈Γ

by inputs xk , k ∈ Γ , subject to technology (4), given the input prices pk , k ∈ Γ , and the output price P . This yields pk = P 6

PC ∂C for k ∈ Γ. = θ(Γ )xk ∂xk

(5)

In other papers, the property that product variety increases welfare is commonly established through a CES production function. In this study, the replacement of the Cobb-Douglas function (4) by a CES function would excessively complicate the analysis.

Common Markets, Economic Growth, and Creative Destruction

61

2.3 Intermediate-Good Producers All potential producers of intermediate good k take part in Bertrand competition and can produce one unit of their product from one labor . unit. They maximize their proﬁt πk = pk xk − wk xk by output xk , given the wage in k ∈ Γ , wk . Only one ﬁrm has access to the technology for a state-of-the-art product, while another is able to manufacture the product that is one step behind on the quality ladder. Because each new generation of any intermediate good provides exactly ε > 1 times as many services as the product of the generation before it, to keep entrants away from the market, the incumbent ﬁrm sets its output price pk equal to εwk . This yields7 . pk = εwk , πk = pk xk − wk xk = (1 − 1/ε)pk xk . (6) Since there is no trade or investment between common markets, we can choose a separate numeraire for each common market Γ and normalize total consumption expenditure P C at θ(Γ ). Given P C = θ(Γ ), (5) and (6), we obtain pk xk = 1,

πk = 1 − 1/ε,

wk = pk /ε = 1/(εxk ).

(7)

2.4 Economic Growth The average productivity of R&D in common market Γ is given by . 1 λk dk, (8) λ(Γ, {λk∈Γ }) = θ(Γ ) k∈Γ where {λk∈Γ } denotes the levels of productivity λk throughout all its members k ∈ Γ . Because the average growth rates are in ﬁxed proportion (log ε) to the arrival rates Λk ,8 we can use Λk as proxies of the growth rates of economies. From the production function (4) it follows that in common market Γ the level of productivity in the consumptiongood sector is given by . ' B= Bk (tk )1/θ(Γ ) , (9) k∈Γ

and the average growth rate of consumption (= the arrival rate of jumps ε > 1 in the level of productivity in the consumption-good sector) by . 1 Λk dk. (10) Λ= θ(Γ ) k∈Γ 7 8

Cf. Grossman and Helpman (1991), chapter 4. For this, see Aghion and Howitt (1998), p. 59.

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2.5 Endowment We assume that there is no migration between economies. There is a ﬁxed number κk of households in each economy k. Each of these κk households supplies 1/κj units of labor, which can be used either in production or in R&D. Total labor devoted to R&D in economy k is . zk = zjk dj, (11) j∈Γ

where zjk is the demand for labor in economy k by R&D ﬁrms located in economy j. Because R&D ﬁrms employ zk units and the intermediategood ﬁrm xk units of labor, and households supply one labor unit in economy k, the equilibrium condition of the labor market is 1 = xk + zk for all k.

(12)

3 Equilibrium Properties Because labor inputs zjk are perfect substitutes in the production function (1), R&D ﬁrms employ labor in the economy k ∈ Γ with the lowest unit cost wk /λk . In equilibrium, wj /λj is then uniform for j ∈ Γ , wj = λj w for j ∈ Γ ,

(13)

where w is the wage per eﬀective labor in R&D in common market Γ . Given (7), (8), (10), (11) and (12), we obtain λk /λj = wk /wj = xj /xk for j, k ∈ Γ , λj xj = λk xk for j, k ∈ Γ , . 1 Λ= θ

k∈Γ

1 Λk dk = θ

k∈Γ

1 λk zk dk = θ

k∈Γ

λk (1 − xk )dk = λ − λj xj .

From (13) and (14) it follows that the demand for labor in production in economy k, xk , and the wage per eﬀective labor w are determined by xj =

λ−Λ , λj

w=

1 1 1 wj for j ∈ Γ , = = ε λ−Λ ελj xj λj

λ > Λ. (14)

We assume that households can save only by investing in the shares of R&D ﬁrms. We denote household ι’s investment in R&D (= ι’s sav(ing) in economy j by Sjι . In economy j, aggregate saving by households, S dι, is equal to the issue of shares by R&D ﬁrms, which in turn is ι jι

Common Markets, Economic Growth, and Creative Destruction

63

( equal to aggregate investment k∈Γ wk zjk dk. Given (1) and (13), this condition takes the form Sjι dι = wk zjk dk = w λk zjk dk = wΛj . (15) ι

k∈Γ

k∈Γ

Let ijι be household ι’s true share of the proﬁts in economy j. The sum of these throughout all households in economy j must be equal to unity: κj ijι dj = 1. (16) ι=1

Household ι ∈ {1, ..., κj } in economy j supplies 1/κj units of labor, and receives the wage wj and the share ijι of the total proﬁt of the intermediate-good ﬁrm, πj .9 Given (13) and (7), its total income is then w w Ajι = κjj + ijι πj = κjj + 1 − 1ε ijι = λj κwj + 1 − 1ε ijι . (17) Because income Ajι is equal to savings Sjι plus consumption expenditure P Cjι , consumption Cjι is determined by Cjι =

1 Sjι 1 w Ajι − Sjι . ijι − + 1− λj = P ε κj P P

(18)

4 Households When a household has ﬁnanced a successful R&D project, it acquires a certain share of the proﬁt the successful ﬁrm earns. We assume that it receives a share of the ﬁrm’s proﬁt which is equal to its investment relative to total investment made in the same successful project. A successful project also implies that the old ﬁrm is driven out of the market. In such a case, all shares held by a household in the old ﬁrm lose their value. Following W¨ alde (1999a), we assume that the change in the true proﬁt share, dijι , is a function of the increment dqj of a Poisson process qj and the investment share Ijι of household ι out of total investment (15): Sjι Sjι . = wΛ S dk j ι∈Γ jι (19)

dijι = (Ijι − ijι )dq for all j and ι, where Ijι = (

9 Because the consumption-good ﬁrms are subject to constant returns to scale, in equilibrium they do not yield proﬁts.

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When a household does not invest at all in the upcoming vintage (i.e., Ijι = 0), its share-holdings are reduced to zero (i.e., dijι = −ijι ) in the case of R&D success dq = 1. If it invests in the upcoming vintage, then the size of the share-holdings depends on relative investment Ijι . A household averts risk and maximizes utility over an inﬁnite horizon, given random technological change. We deﬁne the utility of household ι ∈ {1, ..., κj } in economy j from an inﬁnite stream of consumption beginning at time T as: Uj (Cjι , T ) = E

∞

T

1−β Cjι − 1 −ρj (τ −T ) e dτ, 1−β

ρj > 0,

β < 1,

(20)

where τ is time, E the expectations operator, Cjι consumption by household ι, ρj the constant rate of time preference and β the constant degree of relative risk aversion. Household ι maximizes its utility (20) subject to stochastic process (19) and the budget constraint (18) by its saving Sjι , taking the wages wj and w, the consumption price P and the arrival rate of innovations Λj as given. We denote the value of receiving a share ijι of the proﬁt of the producers possessing current technology tj by Ω(ijι , tj ), and the value of receiving a share Ij of the proﬁt of the producers discovering the next technology tj + 1 by Ω(Ij , tj + 1). The Bellman equation for the household’s program is then10 % 1−β & ρj Ω(ijι , tj ) = max Cjι /(1−β)+Λj [Ω(Ijι , tj +1)−Ω(ijι , tj )] , (21) Sjι ≥0

where Cjι is given by (18). This leads to the ﬁrst-order condition Λj

∂ −β −β ∂Cjι = Cjι /P. [Ω(Ijι , tj + 1) − Ω(ijι , tj )] = −Cjι ∂Sjι ∂Sjι

(22)

We focus entirely on the households’ stationary equilibrium in which the allocation of resources is invariable across technologies, and ignore dynamics during the transitional period before the equilibrium is reached. We try the solution that consumption expenditure P Cjι is a share 0 ≤ cjι ≤ 1 of income Ajι , and that the value function is Ω = (cjι Ajι )1−β /[(1 − β)rjι ],

(23)

where the expenditure-income ratio cjι and the household’s subjective interest rate rjι are independent of income Ajι . Inserting these solutions into (21) and (22), we obtain that in common market Γ the subjective 10

Cf. Dixit and Pindyck (1994), and W¨ alde (1999a), (1999b).

Common Markets, Economic Growth, and Creative Destruction

65

interest rates rjι and the expenditure-income ratios cjι are uniform (Appendix A): . rjι = rj = ρj + [1 − ε(1−β)/θ(Γ ) ]Λj > 0 for all j ∈ Γ , (24) ρj + [1 − ε(1−β)/θ(Γ ) ]Λj Sjι . for all j ∈ Γ . = cjι = cj = min 1, 1− Ajι (ε − 1)(λ − Λ)ε(1−β)/θ(Γ ) (25) Now we can show the following: Proposition 1. A non-integrated economy j does not grow at all, if its households are impatient enough for ρj ≥ (ε − 1)λj ε1−β to hold. Proof: Given (14), (24) and (25), economy j ∈ Γ saves (cj < 1) only if (26) (ε − 1)(λ − Λ)ε(1−β)/θ(Γ ) > ρj + [1 − ε(1−β)/θ(Γ ) ]Λj . If condition (26) does not hold, then economy j ∈ Γ consumes all its income, cj = 1. If this economy is left alone as a separate common market, Γ = {j} and θ = 1, then, given (8), (1), (15) and (25), it does not save or invest in R&D, Sjι = zj = 0, consequently it does not grow at all, Λ = Λj = 0, and λ = λj . Hence, inequality (26) does not hold if and only if ρj ≥ (ε − 1)λj ε1−β ] holds. This means that Λj = 0 for ρj ≥ (ε − 1)λj ε1−β . //// Proposition 1 can be intuitively explained as follows (see Fig. 1 below). In the (xj , zj ) plane, the equilibrium condition of the labor market zj

U0

U1

E

Q1 O

E Q 0

U0

U1

xj

Fig. 1. Possible equilibria for an isolated economy.

66

Tapio Palokangas

(12) corresponds to the downward-sloping line EE. Because economy j must always produce the consumption good (i.e., xj > 0) but need not do R&D (i.e., zj = 0 is possible), preferences in economy j can be represented by indiﬀerence curves which intersect the horizontal axis but approach the vertical axis asymptotically. Depending on preferences, there can then be two cases. When the households are impatient enough (i.e., a high enough rate of time preference, ρj ), there is a corner solution Q0 with no investment and saving, zj = 0, but otherwise there is an interior solution Q1 with investment and saving, zj > 0. The indiﬀerence curve corresponding to the equilibrium Q0 is given by U0 U0 and that corresponding to Q1 by U1 U1 .

5 Economic Integration Because the average growth rate of consumption in common market Γ is given by (10), we obtain our ﬁrst result as follows: Proposition 2. The members of a common market Γ have parallel consumption growth paths. They may have diﬀerent levels of productivity, but their consumption grows at the same rate (10). In contrast to Howitt (2000), where growth convergence results from international technology spillover, the uniﬁcation of the growth rates is here caused by free movement of R&D ﬁrms in the common market. Let Υ be the set of economies which do not save at all. Then, given (25), (27) cj = 1 and Λj = 0 for j ∈ Υ obtains. In Appendix B, we prove Λ Γ, {ρk∈Γ −Υ }, {λk∈Γ } with (ε − 1)[θ(Γ ) − θ(Γ ∩ Υ )] 0< θ(Γ )[εθ(Γ ) − (ε − 1)θ(Γ ∩ Υ )] 1 ∂λ ∂Λ = < < θ(Γ ) ∂λj ∂λj for j ∈ Γ ∩ Υ and a big enough θ(Γ ),

(28)

where {ρk∈Γ −Υ } denotes ρk for k ∈ Γ − Υ and {λk∈Γ } λk for k ∈ Γ . Total consumption in common market Γ is obtained by summing throughout all economies and households in Γ , κ j . C= Cjι dι dj. j∈Γ

0

Common Markets, Economic Growth, and Creative Destruction

67

In the stationary state, given (8), (4), (9), (14) and (28), we can deﬁne ' 1/θ(Γ ) . C = χ Γ, {ρk∈Γ −Υ }, {λk∈Γ } = µ(θ(Γ )) xk B k∈Γ ' −1/θ(Γ ) = µ(θ(Γ )) λ(Γ, {λk∈Γ }) − Λ Γ, {ρk∈Γ −Υ }, {λk∈Γ } λk

k∈Γ

1 1 ∂Λ ∂λ 1 1 ∂χ >− − − = θ(Γ )λj θ(Γ )λj ∂λj λ − Λ ∂λj χ ∂λj for j ∈ Γ ∩ Υ and a big enough θ(Γ ).

with

(29)

Given (8), (20), (28) and (29), we obtain the welfare of the representative household in common market Γ of size θ(Γ ) as follows:

∞

U =E T∞ =E T

1 C 1−β e−ρ(τ −T ) dτ 1−β 1 χ1−β B 1−β e−ρ(τ −T ) dτ. 1−β

(30)

The arrival rate of Poisson jumps in the level of consumption, B, is given by (10). Noting (28), (29) and (30), we obtain the expected value of the stream of welfare (30) subject to the Poisson process as follows: χ(Γ, ρk∈Γ −Υ , λk∈Γ )1−β B(t)1−β (1 − β)[ρ + (1 − ε1−β )Λ(Γ, ρk∈Γ −Υ , λk∈Γ )] ∂Λ ε1−β − 1 1 − β ∂χ 1 ∂Ψ + = with ρ + (1 − ε1−β )Λ ∂λj χ ∂λj Ψ ∂λj 1 ε1−β − 1 β − 1 (ε − 1)[θ(Γ ) − θ(Γ ∩ Υ )] + > εθ(Γ ) − (ε − 1)θ(Γ ∩ Υ ) ρ + (1 − ε1−β )Λ θ(Γ ) λj for j ∈ Γ ∩ Υ and a big enough θ(Γ ). (31)

. Ψ (t, ρk∈Γ −Υ , λk∈Γ ) =

The social planner of common market Γ maximizes the expected welfare Ψ by choosing the members of the common market (i.e., the set Γ ) among those who are willing to join in. To simplify the analysis, we assume that a common market expands by taking new members economy by economy and that an economy can freely leave the common market whenever it wants to do so. We can then prove: Proposition 3. In equilibrium, there can be a number of common markets with diﬀerent positive growth rates as well as a number of stagnating economies which do not grow at all.

68

Tapio Palokangas

Proof: Let 1 be the economy which is most productive in R&D in the whole world, λ1 = maxk λk , and Γ1 the frontier common market containing economy 1. We deﬁne the order of economies in Γ1 as follows. If economy 1 forms the common market alone, Γ1 = {1}, then it prefers economy 2 as the second member, and these two together economy 3 as the third member, etc. Common market Γ1 expands, until any new member j would decrease welfare in Γ1 or any economy j would decrease its welfare by joining in Γ1 : Ψ t, Γ1 , {ρk∈Γ1 −Υ }, {λk∈Γ1 } > Ψ t, Γ1 + {j}, {ρk∈Γ1 −Υ }, ρj , {λk∈Γ1 }, λj or Ψ (t, {j}, ρj , λj ) > Ψ t, Γ1 + {j}, {ρk∈Γ1 −Υ }, ρj , {λk∈Γ1 }, λj . dk. for all j > θ1 = θ(Γ1 ) = Γ1

Let θ1 + 1 be the economy which is the most productive in R&D among those economies which are outside Γ1 , and Γ2 the common market containing economy θ1 + 1. By the same reasoning as above, we obtain the following equilibrium condition for Γ2 : Ψ t, Γ2 , {ρk∈Γ2 −Υ }, {λk∈Γ2 } > Ψ t, Γ2 + {j}, {ρk∈Γ2 −Υ }, ρj , {λk∈Γ2 }, λj or Ψ (t, {j}, ρj , λj ) > Ψ t, Γ2 + {j}, {ρk∈Γ2 −Υ }, ρj , {λk∈Γ2 }, λj ( . for all j > θ2 = θ(Γ2 ) = Γ2 dk. Hence, by induction, we obtain a (ﬁnite) sequence {Γi } of common markets. Outside sets {Γi }, there are only economies which do not grow at all. //// Proposition 3 can be intuitively explained as follows (see Fig. 2, page 69). Given (29), there is a negative relationship E0 E0 between C and Λ: the more resources are devoted to R&D (i.e., the higher Λ), the less common market Γ can consume (i.e., the lower C). The utility function (31) can be represented by a family of indiﬀerence curves which approach the vertical axis asymptotically but may intersect the horizontal axis. Let the original equilibrium be Q0 and the corresponding indiﬀerence curve be U0 U0 . The adoption of a new member in common market Γ has two eﬀects. First, with a wider variety of products, the productivity of the consumption-good sector increases and the resource constraint shifts to the right E0 E0 → E0 E1 .11 Second, if the new member has a lower level of productivity in R&D than the remainder of 11

E0 .

If there is no consumption, C = 0, then these two curves coincide at point

Common Markets, Economic Growth, and Creative Destruction

69

Λ

U0

U1

E0 E2

Q 0 Q1

U1 O

E0

U0

E

1

C

Fig. 2. The expansion of a common market.

common market Γ , then the arrival rate of innovations, Λ, falls. This shifts the resource constraint to the left E0 E1 → E2 E1 .12 As long as the former eﬀect is stronger, the new equilibrium Q1 is above U0 U0 and welfare in common market Γ increases U0 U0 → U1 U1 with the new member.13 Each new member is less and less productive in R&D, which means that the shift E0 E1 → E2 E1 increases with the number of members. When there are enough members in common market Γ , the new equilibrium is is below U0 U0 and no more applicants will be accepted. This shows that the optimal size of common market Γ can be ﬁnite. Consequently, there can be many common markets which grow at diﬀerent rates. Some economies can form common markets alone which do not grow at all. Our last result is the following: Proposition 4. (Windows of opportunity) Consider economy j ∈ Υ which has insuﬃcient saving incentives (i.e., with a too high ρj ) to grow if left alone. This economy can grow if the productivity of its R&D, λj , is so high that it can join a large common market with a positive growth rate. 12

If there is no R&D, Λ = 0, then these two curves coincide at point E1 . If the new member has a higher level of productivity in R&D than the remainder of common market Γ , then the arrival rate of innovations, Λ, rises, and the resource constraint shifts further to the right. In this case, the applicant is accepted as a member right away. 13

70

Tapio Palokangas

Proof: Given proposition 1 and (27), inequality ρj ≥ (ε − 1)λj ε1−β holds for economy j ∈ Υ . We assume that economy j has an incentive to join in Γ , i.e., its welfare is higher as a member of Γ than alone: Ψ (t, {j}, ρj , λj ) ≤ Ψ t, Γ + {j}, {ρk∈Γ −Υ }, ρj , {λk∈Γ }, λj . Common market Γ accepts economy j as a member, if Ψ t, Γ, {ρk∈Γ −Υ }, {λk∈Γ } ≤ Ψ t, Γ + {j}, {ρk∈Γ −Υ }, ρj , {λk∈Γ }, λj . (32) Now assume that common market Γ is large enough (i.e., θ(Γ ) is big enough). Given (31), we then obtain ∂Ψ/∂λj > 0 for λj > (1 − β)

ρ ε1−β − 1

−Λ

εθ(Γ ) − (ε − 1)θ(Γ ∩ Υ ) . (ε − 1)[θ(Γ ) − θ(Γ ∩ Υ )]

From this and (32) it follows that if economy λj is large enough, then common market Γ accepts it as a member. Combining the two inequalities, we conclude that when common market Γ is large enough and the productivity of R&D in economy j, λj , is within the range (1 − β)

ρ ε1−β − 1

−Λ

εθ(Γ ) − (ε − 1)θ(Γ ∩ Υ ) ρj , < λj ≤ (ε − 1)ε1−β (ε − 1)[θ(Γ ) − θ(Γ ∩ Υ )]

then economy j does not save and grow at all if left alone, but starts growing if it joins Γ . //// Proposition 4 can be explained as follows (see Fig. 3, page 71). Let U0 U0 be the indiﬀerence curve for the representative household, E0 E0 the resource constraint, Q0 the equilibrium in economy j which alone forms a common market but which does not grow at all,14 and U1 U1 the indiﬀerence curve for the representative household in common market Γ as regards to the use of labor in economy j. On the other hand, let U1 U1 be the indiﬀerence curve for the representative household, E1 E1 the resource constraint, and Q1 the equilibrium in common market Γ . If economy j is very small relative to common market Γ , then the preferences U1 U1 , the resource constraint E1 E1 and the equilibrium Q1 will not signiﬁcantly change with the membership of economy j. If Q1 is above U0 U0 , economy j applies the membership of Γ and it is accepted as a member if its productivity in R&D is high enough. Because Q1 is above the horizontal axis and (in line with proposition 2)) all members of Γ grow at a positive rate, the membership of Γ starts investment and technological change in economy j. 14 Curve E0 E0 is a line, because it corresponds to equation (12) in the case the economy is isolated.

Common Markets, Economic Growth, and Creative Destruction

71

Λ U

E

1

U0

Q1

U O

Q0

E U

1 C

0

Fig. 3. Windows of opportunity.

6 Conclusions This paper examines a multi-economy world where growth is generated by creative destruction as follows. Consumption is produced from intermediate goods. An R&D ﬁrm that creats the latest technology of an intermediate good through a successful R&D project crowds the other ﬁrms with older technologies out of the market so that they lose their value. The ﬁrms ﬁnance their R&D by selling shares, and households save only by buying these shares. Any set of economies can form a common market in which intermediate goods are freely traded and R&D ﬁrms can freely operate. We examine the eﬀects of the establishment and expansion of a common market on growth and social welfare. The main ﬁndings can be summarized as follows. Common markets that do more R&D, produce more innovations and grow faster. The expansion of a common market has two eﬀects. New members increase welfare through a wider variety of products. On the other hand, if the new members have a lower productivity of R&D or a lower savings rate than the old members of the common market, technological change will be slower and the level of welfare lower for the old members. When these advantages and disadvantages of economic integration are balanced, the common market is of optimal size. Economies integrate only if the productivity of R&D does not diﬀer too much for them. After integration they grow at the same rate, but at diﬀerent levels of productivity. The integration of two common markets

72

Tapio Palokangas

increases the growth rate of the one whose R&D is less productive and decreases that of the other whose R&D is more productive. If the productivity of R&D diﬀers too much for them, then the one with the less productive R&D declines the integration. For it, the welfare loss of a lower growth rate outweighs the welfare beneﬁt of a wider variety of products. Because economies with roughly the same productivity of R&D integrate, there can be a number of common markets with diﬀerent growth rates. Some economies stagnate, because their incentives to save are too weak (e.g., households are too impatient) for R&D and growth. Such an economy can, however, escape from stagnation if it is productive enough in R&D to join a large common market with persistent growth. It can then grow even without domestic saving, because its R&D projects will be wholly ﬁnanced by the savings of the other member economies.

Appendix A. The Behavior of a Household We denote variables depending on technology tj by superscript tj . t t Since, given (17), household ι’s income Ajιj depends on ijιj , we note tj tj Ajι (ijι ). On the assumption that cj is invariable across technologies, we obtain t

t

t

t

t

t

t

P tj Cjιj = cj Ajιj (ijιj ), Sjιj = (1−cj )Ajιj (ijιj ) = (1/cj −1)P tj Cjιj . (A. 1) Household ι’s share in the next producer tj + 1 is determined by its t +1 t investment under technology tj , ijιj = Ijιj . Given (23), the value functions are

t

t

t

t

Ω(ijιj , tj ) =

(Cjιj )1−β (1−β)rj

=

t +1

t +1

Ω(Ijιj , tj + 1) =

(Cjιj )1−β (1−β)rj

t

[cj Ajιj (ijιj )/P tj ]1−β , (1−β)rj

=

[cj Ajιj

t

(Ijιj )/P tj +1 ]1−β . (1−β)rj

(A.2)

Given this and (17), we obtain t

t

∂Ω(ijιj , tj )/∂Sjιj = 0.

(A.3)

Common Markets, Economic Growth, and Creative Destruction

73

From (1), (8), (19), (17) and (A.2) it follows that t

∂Ijιj

t ∂Sjj

=

t +1

1 wtj Λ

t ∂Ω(Ijιj , tj t ∂Sjj

∂[Ajιj

tj ,

t ∂Ijιj

j

+ 1)

t +1

t

(Ijιj )]

=

t

= (1 − β)

∂[Ajιj

Ω(Ijιj , tj + 1) t +1

Ajιj

t +1

(ijιj

t +1 ∂ijιj t +1 ∂Ajιj t ∂Ij j

)]

1 =1− , ε t

∂Ij j t

∂Sjj

Ω(Ijιj , tj + 1) 1 (1 − β) tj +1 = 1− t . ε Ajι wtj Λjj

t

(A.4)

We focus on a stationary equilibrium in which the allocation of t t labor in (12) is invariable across technologies, zjj = zj and xjj = xj . Given this, (1), (7), (17) and (A. 1), wages, income, expenditure and the arrival of innovations are also invariable across technologies, wtj = w, t t t +1 t Ajιj = Ajι , P tj Cjιj = P tj +1 Cjιj and Λjj = Λj . Because in a stationary t state consumption Cjιj and the level of productivity in the consumptiontj good sector, B , grow at the same rate, given (9) and (A.2), we obtain t +1

P tj /P tj +1 = Cjιj

t

/Cjιj = B tj +1 /B tj = [Bj (tj + 1)/Bj (tj )]1/θ(Γ )

= ε1/θ(Γ ) , t Ω(Ijιj , tj

t

t +1

+ 1)/Ω(ijιj , t) = (Cjιj

t

/Cjιj )1−β = ε(1−β)/θ(Γ ) .

(A.5)

Inserting these results, (A. 1) and (A.2) into equation (21), we obtain t

t

t

0 = (ρj + Λj )Ω(ijιj , t) − (Cjιj )1−β /(1 − β) − Λj Ω(Ijιj , tj + 1) ) * t = ρj − rj + [1 − ε(1−β)/θ(Γ ) ]Λj Ω(ijιj , t). This leads to (24). Inserting (14), (24) and (A. 1)-(A.5) into (22) and noting (7) yield t t t (Cjιj )−β ∂Ω(Ijιj , tj + 1) ∂Ω(ijj , t) − − 0 = Λj t t P tj ∂Sjj ∂Sjj t

t

= (1 − 1/ε)(1 − β)Ω(Ijιj , tj + 1)/(Ajι w) − (Cjιj )−β /P tj t

t

= (ε − 1)(λ − Λ)(1 − β)Ω(Ijιj , tj + 1)/Ajι − (Cjιj )−β /P tj t (C tj )−β P tj Ω(Ijιj , tj + 1) jι − 1 = (ε − 1)(λ − Λ)(1 − β) t P tj Ajι (Cjιj )−β t (C tj )−β P tj Ω(ijιj , t) jι (1−β)/θ(Γ ) −1 = (ε − 1)(λ − Λ)(1 − β)ε t P tj Ajι (Cjιj )−β

74

Tapio Palokangas t (C tj )−β P tj Cjιj jι −1 = (ε − 1)(λ − Λ)ε(1−β)/θ(Γ ) P tj Ajι rj (C tj )−β cj jι = (ε − 1)(λ − Λ)ε(1−β)/θ(Γ ) − 1 P tj rj t (Cjιj )−β (ε − 1)(λ − Λ)ε(1−β)/θ(Γ ) cj . −1 = P tj ρj + [1 − ε(1−β)/θ(Γ ) ]Λj

This, (24) and cj < 1 yield (25), the expenditure-income ratio. B. The Function (28) Given (3), (8), (10), (15), (16), (14), (17), (25) and (27), we obtain 1 Sjι dι = (1 − cj )Aj /w = (1 − cj )[λj + (1 − 1/ε)/w] Λj = w ι % & = (1 − cj ) λj + (ε − 1)[λ(Γ, {λk∈Γ }) − Λ] for j ∈ Γ − Υ , (A.6) cj = . Λ = = = =

=

=

ρj + [1 − ε(1−β)/θ(Γ ) ]Λj for j ∈ Γ − Υ , (A.7) (ε − 1)[λ(Γ, {λk∈Γ }) − Λ]ε(1−β)/θ(Γ ) 1 1 Λk dk Λk dk = θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ 1 (1 − ck )[λk + (ε − 1)(λ − Λ)]dk θ(Γ ) k∈Γ −Υ λ−Λ 1 (1 − ck )dk (1 − ck )λk dk + (ε − 1) θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ −Υ 1 1 ck λk dk λk dk − θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ −Υ

1 1 ck dk dk − +(ε − 1)(λ − Λ) θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ −Υ 1 1 1 ck λk dk λk dk − λk dk − θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ ∩Υ θ(Γ ) k∈Γ +(ε − 1)(λ − Λ)

1 1 1 ck dk dk − dk − θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ ∩Υ θ(Γ ) k∈Γ 1 1 ck λk dk λk dk − λ(Γ, {λk∈Γ }) − θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ ∩Υ +(ε − 1)[λ(Γ, {λk∈Γ }) − Λ]

1 θ(Γ ∩ Υ ) ck dk . (A.8) − 1− θ(Γ ) k∈Γ −Υ θ(Γ )

Common Markets, Economic Growth, and Creative Destruction

75

In the system (A.6), (A.7) and (A.8), there are 2θ(Γ ) − 2θ(Υ ) + 1 equations, 2θ(Γ ) − 2θ(Υ ) + 1 endogenous variables, Λj for j ∈ Γ − Υ , cj for j ∈ Γ − Υ , and Λ, and exogenous variables ρj for j ∈ Γ and λj for j ∈ Γ − Υ . This system deﬁnes the function Λ in (28). Letting θ(Γ ) → ∞ for (A.7) and (A.8), we obtain ε(1−β)/θ(Γ ) → 1, and

cj = ρj /[(ε − 1)(λ − Λ)]

1 ck λk dk λk dk − θ(Γ ) k∈Γ −Υ k∈Γ ∩Υ

1 θ(Γ ∩ Υ ) ck dk − +(ε − 1)(λ − Λ) 1 − θ(Γ ) k∈Γ −Υ θ(Γ ) 1 1 ρk λk dk λk dk − = − (ε − 1)(λ − Λ)θ(Γ ) k∈Γ −Υ θ(Γ ) k∈Γ ∩Υ

1 θ(Γ ∩ Υ ) ρk dk. (A.9) − +(λ − Λ) ε − (ε − 1) θ(Γ ) k∈Γ −Υ θ(Γ )

1 0 = λ−Λ− θ(Γ )

From (3) and (8) it follows that ∂λ/∂λj = 1/θ(Γ ) and θ(Γ ∩ Υ ) < θ(Γ ). Noting these results and ε > 1, and diﬀerentiating the equation (A.9) with respect to λj for j ∈ Γ ∩ Υ , we obtain 1 1 ρk λk dk = (ε − 1)(λ − Λ)2 θ(Γ ) k∈Γ −Υ θ(Γ ) .+ ,+ . ,+

+

∂Λ ∂λ θ(Γ ∩ Υ ) − + ε − (ε − 1) ∂λj ∂λj θ(Γ ) . + ,+ . ,

+

∂Λ ∂λ θ(Γ ∩ Υ ) − > ε − (ε − 1) ∂λj ∂λj θ(Γ )

∂Λ 1 θ(Γ ∩ Υ ) − = ε − (ε − 1) θ(Γ ) ∂λj θ(Γ )

+

(A.10)

and 0<

1 ∂λ ∂Λ (ε − 1)[θ(Γ ) − θ(Γ ∩ Υ )] . = < < θ(Γ ) ∂λj ∂λj θ(Γ )[εθ(Γ ) − (ε − 1)θ(Γ ∩ Υ )]

76

Tapio Palokangas

Acknowledgements The author thanks the participants in the eighth international conference on “Dynamics, Economic Growth and International Trade (DEGIT)”, Helsinki, Finland, May 30-31, 2003, and two anonymous referees for useful comments. Financial support from the Yrj¨ o Jahnsson Foundation is gratefully acknowledged. Needless to say, the author is responsible for any remaining errors.

References Aghion, P., and Howitt, P. (1998): Endogenous Growth Theory. Cambridge, MA: MIT Press. Baldwin, R.E., and Forslid, R. (2000): “Trade Liberalization and Endogenous Growth: a q-Theory Approach.” Journal of International Economics 50: 497–517. Barro, R.J. (1998): Determinants of Economic Growth. Cambridge, MA: MIT Press. Barro, R.J., and Sala-i-Martin, X. (1994): Economic Growth. New York: McGraw-Hill. Dixit, A., and Pindyck, K. (1994): Investment under Uncertainty. Princeton: Princeton University Press. Grossman, G., and Helpman, E. (1991): Innovation and Growth. Cambridge, MA: MIT Press. Howitt, P. (2000): “Endogenous Growth and Cross Country Diﬀerences.” American Economic Review 90: 829–846. Peretto, P.F. (2003): “Endogenous Market Structure and the Growth and Welfare Eﬀects of Economic Integration.” Journal of International Economics 60: 177–201. Ventura, J. (1997): “Growth and Interdependence.” The Quarterly Journal of Economics 112: 57–84. Walz, U. (1998): “Does an Enlargement of a Common Market Stimulate Growth and Convergence.” Journal of International Economics 45: 297–321. W¨ alde, K. (1999a): “A Model of Creative Destruction with Undiversiﬁable Risk and Optimizing Households.” The Economic Journal 109: C156–C171. W¨ alde, K. (1999b): “Optimal Saving under Poisson Uncertainty.” Journal of Economic Theory 87: 194–217. Address of author: – Tapio Palokangas, Department of Economics, P.O. Box 17 (Arkadiankatu 17), 00014 University of Helsinki, Finland (e-mail: tapio.palokangas@helsinki.ﬁ)

J. Econ. (2005) Suppl. 10: 77-118

Schumpeterian Growth and the Political Economy of Employment Protection Wolf-Heimo Grieben Received September 2, 2003; Revised version received June 17, 2004 c Springer-Verlag 2005

This paper analyzes the diﬀering attitudes concerning political support for employment protection between skilled and unskilled workers in a qualityladder growth model. Creative destruction through innovation results in “Schumpeterian unemployment” of unskilled workers. By voting on ﬁring costs, unskilled workers consider a trade-oﬀ between the beneﬁt of fewer unemployment spells and the cost of lower quality growth of consumer goods. Skilled workers, although not threatened by unemployment, may vote for even larger ﬁring costs. Alleviating one labor market rigidity by increasing the matching eﬃciency between ﬁrms and unskilled workers aggravates another rigidity by creating political support for additional ﬁring costs. Keywords: Non-Scale Growth, Schumpeterian Unemployment, Firing Costs JEL classiﬁcation: J63, O33, E24, D72

1 Introduction Employment protection is often considered to be excessive in many European countries, resulting in involuntary unemployment.1 More speciﬁcally, ﬁring costs are set to maintain workers in dying industries. Instead of simply advising politicians to remove these ﬁring costs, serious economists need to understand the underlying political economics of employment protection legislation. This seems to be even more important since rigid European labor markets are often claimed to cause 1 The OECD (1999, Chapt. 2) provides an extensive summary of employment protection legislation (EPL) for 27 OECD countries and analyses its eﬀects on employment. No signiﬁcant eﬀect of the strictness of EPL on overall unemployment is found. However, the problem of long-term unemployment tends to be aggravated by stricter EPL because job turnover is dampened signiﬁcantly.

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a long-run growth slowdown in addition to creating unemployment. Therefore, this paper aims at analyzing the driving forces behind the demand for employment protection of diﬀerent groups of workers within a simple and by now standard Schumpeterian endogenous growth model. As argued in a stimulating paper by Saint-Paul (2002) within a vintage model of exogenous growth, incumbent workers form a majority and vote for a speciﬁc level of employment protection by considering the trade-oﬀ between the costs and beneﬁts of it. The beneﬁts are due to a longer job duration at a wage strictly above the alternative income set by the social security net (opportunity costs of working). The diﬀerence between the wage rate and the opportunity costs of working is a rent that exists because of the assumption of match-speciﬁc human capital - workers of a speciﬁc vintage have larger productivity than unemployed. The costs of employment protection consist of staying in a less productive ﬁrm at a wage below the level that could be paid in the leading-technology ﬁrm. Saint-Paul shows that in his model, these costs increase with the growth rate (rate of creative destruction), thereby reducing the political support for ﬁring costs. Firing costs are set according to the single-peaked preferences of the median (employed) voter. Our growth framework is borrowed from the seminal paper of Dinopoulos and Segerstrom (1999). They build a two-country Schumpeterian quality ladder growth model without scale eﬀects that is used to analyze the eﬀects of North-North trade liberalization on wage inequality. More precisely, we apply the PEG (“permanent eﬀects on growth”) version of that model, which results in a steady-state innovation and growth rate that can be inﬂuenced by public policies. The basic quality-ladder framework builds on the model of Grossman and Helpman (1991), where consumers only buy goods with the lowest quality-adjusted price.2 We will restrict the analysis to a closed economy, but introduce labor market frictions. This is done through the use of S ¸ ener’s (2001) matching model of “Schumpeterian unemployment”, where the ﬂow into the pool of unemployed workers is determined by the endogenous innovation rate that forces technologically-backward ﬁrms to shut down and lay oﬀ their (unskilled) production workers. In that setting, jobs are created by new innovative ﬁrms, and R&D is only performed by skilled workers who do not face the risk of becoming unemployed. However, a successful match between the technology of a new quality leader and an unemployed worker only takes place after a search period of given exogenous length, which results in matching unemployment. Contrary to the model of S¸ener, the exit of old quality leaders overtaken by 2 Other pioneering contributions to neo-Schumpeterian growth theory include most notably Segerstrom et al. (1990) and Aghion and Howitt (1992).

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new innovators is costly in our model, and all workers as well as the unemployed vote on the level of ﬁring costs. Due to the assumption of perfect unemployment insurance, unskilled employed and unemployed workers have the same preferences. Contrary to the model of Saint-Paul (2002), we have no further need for the assumption of a productivity diﬀerential between employed and unemployed workers that creates the rent for incumbent workers, which in turn provides incentives to vote for employment protection. Here, productivities of employed and unemployed workers are identical. However, the time spent for a successful match with an unemployed worker involves costs for an incumbent ﬁrm (forgone proﬁt during search period), which results in an unskilled wage rate strictly above the level of unemployment beneﬁts. The steady-state unemployment rate of unskilled workers increases in proportion to the (endogenous) innovation rate. An increase in ﬁring costs is shown to reduce the steady-state innovation rate and to increase the proportion of unskilled workers. The latter follows from the declining unemployment rate which raises expected wage earnings of unskilled workers, thereby reducing education incentives. We also derive a critical level of the innovation rate below (above) which an increase in ﬁring costs raises (reduces) the skilled wage rate. We assign empirically ‘plausible’ values to the parameters of our model and ﬁnd that for most of the parameter ranges, the innovation rate will stay below this critical level. Hence in most cases, after a change in ﬁring costs, the steady-state innovation rate and the skilled wage rate move in opposite directions. When considering a vote for a speciﬁc level of ﬁring costs, both worker groups take into account a trade-oﬀ between the beneﬁt of rising per capita consumption quantity (due to a higher expected wage income) and the cost of a declining growth rate of consumer goods quality, both in terms of the individual discounted lifetime utility of a given household. Workers do not have to take into account the eﬀects on ﬁrm proﬁts, because we assume (as does Saint-Paul, 2002) that workers do not own ﬁrms, and proﬁts go to a small group of capitalists whose political power can be neglected for the sake of simplicity. This will allow us to concentrate on the political conﬂict between skilled and unskilled workers, instead of mixing it up with a conﬂict between shareholders and workers. However, we will also discuss how our results are aﬀected when accounting for workers’ ﬁrm ownership. The unique preferred level of ﬁring costs is explicitly derived for the unskilled. It depends positively on the (exogenous) eﬃciency of the matching function. Since a more eﬃcient matching technology is shown to increase the steady-state innovation and growth rate, we can conclude the following: in the case of a more ﬂexible labor market with

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a higher turnover that accelerates economic growth, unskilled workers ceteris paribus favor stricter employment protection. If this vote turns into actual economic policy, it slows down product quality growth but reduces unskilled unemployment. The latter raises expected lifetime earnings and thereby increases consumption quantity of any unskilled worker. Furthermore, if the innovation rate is below the critical level, skilled workers would favor even stricter employment protection of the unskilled workers than the unskilled themselves (because of the positive eﬀect on the skilled wage rate).3 Our results shed light on the questions of whether a high-growth or a low-growth environment tends to create more support for policies to make the labor market more ﬂexible, and whether cross-country growth diﬀerentials can be attributed to diﬀerences in speciﬁc aspects of labor market ﬂexibility. The remainder of this paper is organized as follows: Sect. 2 presents the building blocks of our model, which comprises household and ﬁrm behavior as well as a description of how unemployment is created in our model. Sect. 3 derives the unique steady-state equilibrium of our model with endogenous growth and a positive unemployment rate of unskilled workers. In addition, we analyze in detail the various steadystate eﬀects of rising ﬁring costs. Sect. 4 derives the preferences for ﬁring costs of skilled and unskilled workers. Sect. 5 discusses the interaction of diﬀerent aspects of labor market ﬂexibility and the trade-oﬀ between growth and employment protection. It is established that the link between labor market ﬂexibility and growth may be more complex than is usually supposed. In addition, we place our ﬁndings in the context of important contributions of Saint-Paul (2002) and Arnold (2002) on this topic. In Sect. 6, after discussing the results and possible limitations, our analysis is extended in several dimensions. In particular, we cover the case of a net negative eﬀect of ﬁring costs on hiring, and we discuss the case of repeated voting on ﬁring costs. Sect. 7 provides our conclusions.

2 The Model 2.1 Household Behavior and Skill Acquisition The household side of the model is almost identical to Dinopoulos and Segerstrom (1999) and repeated by S ¸ ener (2001), hence we can be brief. 3 If public opinion polls revealed a stronger support for strict EPL among less-skilled than among skilled workers, this must not necessarily be viewed as empirical evidence against our theoretical results. Actual voting behavior may be short-sighted, or voters may lack rational expectations. Our results refer to the case of fully rational workers as members of dynastic families with a bequest motive who vote on the level of ﬁring costs for an inﬁnite horizon.

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There is a continuum of households indexed by ability θ ∈ [0, 1] with all members of household θ having the same ability level equal to θ that is common knowledge for the individual member himself and all potential employers. Furthermore, all households have the same number of members at each point in time, and the total population is growing exogenously according to N (t) = N0 ent with normalization N0 ≡ 1. The optimization problem of a dynastic family with ability θ and a constant subjective time preference rate ρ > 0 is to maximize its discounted utility ∞ Zθ = e−(ρ−n)s log zθ (s)ds (1) t=0

by choice of the allocation of labor income across consumer goods that enter the instantaneous utility function zθ (s) at each point in time s, by the evolution of consumption expenditure over time, and by the decision whether to become skilled or not. There is a continuum of vertically diﬀerentiated goods Q, each produced at a single industry out of a continuum ω ∈ [0, 1]. We denote by qθ (j, ω, s) the quantity of vertically diﬀerentiated goods with j improvements of its quality (each improvement being of size λ > 1) in industry ω at time s consumed by an individual with ability θ. Since the goods produced in each industry just diﬀer in their quality, and λ units of quality j are a perfect substitute for one unit of quality j + 1, only goods with the lowest quality-adjusted price p are consumed. As is usual in quality-ladder growth models (see Grossman and Helpman, 1991), it follows that there is a unit-elastic demand function qθ (j, ω, s) = c(s, θ)/p in each industry ω, where c(s, θ) is the consumption expenditure per capita. Thus, we can deﬁne the instantaneous utility function of each household member as 1 c(s, θ) dω. (2) log zθ (s) = log λj(ω,s) p 0 Utility maximization is subject to an intertemporal budget constraint ∞ Wθ (t) = ens e−[R(s)−R(t)] c(s, θ)ds (3) t

where Wθ (t) denotes the family’s discounted wage income from time t (t ˙ = r(t) on, and R(t) ≡ 0 r(s)ds is the market discount factor with R(t) denoting the instantaneous interest rate at time t. Note that we do not need to rule out savings altogether. Households can borrow and lend with each other or with the capitalists at the market interest rate. The only necessary additional assumption in (3) is that we do not allow workers to own ﬁrms (ﬁrm ownership by the capitalists yields revenues in this type of model because quality leaders earn monopoly proﬁts).

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Therefore, the value of ﬁrms - that will be aﬀected by the level of ﬁring costs - does not aﬀect the workers’ welfare. As is usual, the steady-state interest rate is determined by the optimal saving decision of workers and capitalists (where in the capitalists’ problem, the accumulation of ﬁnancial assets is accounted for in the budget constraint replacing (3)). This yields the usual Keynes-Ramsey rule c/c ˙ = rt − ρ for the optimal path of per-capita consumption expenditure of workers and capitalists. In a steady state with a constant nominal wage rate, per-capita consumption of all individuals must be constant, which implies rt = ρ. We assume ρ > n to have ﬁnite lifetime utility for all households. Each individual lives for an exogenously given period of time D > 0. He can choose either to remain unskilled for his entire lifetime, earning the wage rate wL regardless of his ability if employed or zero if unemployed (unemployment beneﬁts are normalized to zero), or to spend an exogenously given period T < D on education in order to become a skilled worker. This second possibility yields zero income during the period of education and an ability-dependent wage income θwH during the period of employment of length D − T without any unemployment risk. Therefore, an individual with ability θ invests in education if and only if D D −ρt e wL (s)(1 − u)ds < e−ρt θwH (s)ds (4) 0

T

holds.4 Setting (4) to hold as an equality and normalizing wL ≡ 1, we get the unique steady-state threshold ability level as θ0 =

1−u (1 − u)wL 1 − e−ρD σ, σ > 1. ≡ −ρT −ρD wH e −e wH

(5)

Individuals with ability below θ0 ∈]0, 1[ choose to remain unskilled whereas those with ability θ ≥ θ0 invest in education. 2.2 Production and R&D The production side of the model is a closed-economy version of S ¸ ener (2001). In any industry ω ∈ [0, 1], the production function is simply Qt = (1 − u)Lt = (1 − u)θ0 Nt , irrespective of the corresponding quality level. Hence, output equals the amount of employed unskilled labor, 4 Writing the unskilled discounted wage income this way assumes implicitly the existence of perfect unemployment insurance - these workers only care about the expected lifetime income, not about length or frequency of unemployment spells in particular, and the wage income (1 − u)wL is certain. This will simplify our analysis later on since it means identical policy preferences for unskilled employed and unemployed workers.

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where Lt = θ0 Nt is the unskilled labor supply as a fraction θ0 of total population. The R&D process speciﬁed below results in a unique quality leader in each industry who is protected by an exclusive patent on his production technology. Once another innovation (that is, an improvement of the quality of consumer goods of size λ > 1) occurs in the same industry, this patent expires and becomes common knowledge. The current quality leader in each industry reaps all proﬁts by engaging in limit pricing: Since consumers only buy goods with the lowest quality-adjusted price, the quality leader charges the price p = λ−, → 0, that drives all quality followers (who produce one-step inferior goods and can do no better than charging the unit-cost price 1) out of the market. Given a unitelastic demand function q = c/λ, marginal production costs wL = 1 and a total of N consumers, this strategy yields monopoly proﬁts of π = (λ − 1)cN/λ. Total consumption cN/λ equals total output Q and thus total unskilled labor demand, hence it follows: c = λ(1 − u)θ0 .

(6)

This establishes a one-to-one relationship between per-capita consumption quantity q = c/λ and the employed unskilled workforce. There are sequential and stochastic R&D races in each industry ω ∈ [0, 1]. There is free entry in R&D, and all participating ﬁrms are using the same technology in discovering the next higher quality of consumer goods. Skilled labor is the only input used to carry out R&D in any industry ω and is assumed to be perfectly mobile across industries. Any R&D ﬁrm i that hires hi units of skilled labor in industry ω at time t is successful in discovering the next higher quality product with instantaneous probability ηhi /X(ω, t), where X(ω, t) is an R&D diﬃculty index and η > 0 is a given technology parameter. Therefore, ηhi /X(ω, t) “[...] is the probability that the ﬁrm will innovate by time t + dt conditional on not having innovated by time t, where dt is an inﬁnitesimal increment of time” (Segerstrom, 1998, p. 1297). Since we assume that R&D returns are independently distributed across ﬁrms, across industries and over time, the industry-wide instantaneous probability of innovation at time t is I(ω, t) = ηH(ω, t)/X(ω, t)

(7)

with i hi (ω) = H(ω) as the industry-wide employment of skilled labor in R&D.5 R&D diﬃculty is assumed to be proportional to the size of 5 This means that the arrival of innovations in each industry ω is governed by a Poisson process with arrival rate I(ω, t) which in turn depends on R&D employment in a linear way.

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the market:6 X(ω, t) = kNt , k > 0.

(8)

Next, we will consider the stock market. Usually in this type of model, consumer savings are lent to ﬁrms investing in R&D. However, since we will have consumers voting on ﬁring costs which ultimately affects unemployment, the skilled wage rate, the education decision and growth, we want to avoid considering eﬀects on share values simultaneously. Therefore, we follow Saint-Paul (2002) in assuming that savings in R&D funds (however, not savings altogether) come from a small class of capitalists who do not play a signiﬁcant role in the political process to be discussed later. In Sect. 6.1 we will argue that this assumption does not aﬀect our main results qualitatively. As is standard in quality-ladder growth models, eﬃciency in the stock market requires that the expected rate of return of a stock issued by a successful R&D ﬁrm must be equal to the riskless rate of return r (since capitalists can completely diversify the industry-speciﬁc risk of unsuccessful R&D expenditure by holding shares of all existing ﬁrms). Deﬁning ϑ(ω, t) as the expected discounted proﬁts of a monopolist, the no-arbitrage condition in the stock market states ϑ+F ϑ˙ π Idt = rdt. (9) dt + dt(1 − Idt) − ϑ ϑ ϑ The ﬁrst term on the LHS is the dividend per share during the time interval dt, the second term on the LHS denotes share appreciation during this time interval (due to increasing consumption demand because of population growth). This only takes place when there is no further innovation in this industry within dt (the probability of this is given in brackets). The third term on the LHS denotes full capital loss for share owners of this particular ﬁrm when there is further innovation during dt which is often called the “creative-destruction eﬀect”. In addition, this term contains the new variable Ft ≡ BNt , B > 0, denoting ﬁring costs that are indexed to the size of the economy, and that the previous quality leader has to bear in case innovation by a competitor drives him out of the market. It is assumed that Ft does not include redundancy payments to the laid-oﬀ worker. The indexing is necessary in order to 6

Dinopoulos and Segerstrom (1999, p. 459) call this the “permanenteﬀects-on-growth”(PEG) speciﬁcation, because it implies a steady-state innovation and growth rate that can permanently be aﬀected by various economic policies. The intuition behind this speciﬁcation is that the larger the market, the higher are marketing costs to introduce a new technology and the higher are the replacement costs in the case of ﬂaws in newly invented products (Dinopoulos and Thompson, 1996, p. 399). The main purpose of rising R&D diﬃculty over time is to remove counterfactual scale eﬀects from the growth model. See Jones (1995, 1999) and Segerstrom (1998) for detailed discussions of scale eﬀects and diﬀerent ways to avoid them.

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prevent ﬁring costs becoming negligible in a growing economy. Later on, we will introduce the voting of all workers on B. All three terms on the LHS of (9) together denote expected returns per share of any ﬁrm investing in R&D during dt, and this must equal the return r of a riskless asset during the same time period given on the RHS. In the limit dt → 0, we obtain ϑ(ω, t) =

π(t) − F I(ω, t) r(t) + I(ω, t) −

˙ ϑ(ω,t) ϑ(ω,t)

,

(10)

which gives us the appropriately discounted “reward for innovating” with π = (λ − 1)cN/λ. In order not to discourage innovation activity completely, ﬁring costs have to satisfy BI <

λ−1 c. λ

(11)

Note that (10) gives the reward for innovating only if the ﬁrm immediately starts producing after having innovated the leading technology. However, following S ¸ ener (2001), we assume a frictional labor market for unskilled workers which requires that the new quality leader searches for an exogenously given time period y > 0 for the appropriate unskilled worker to be matched with the new vacancy. Thus, given the interest rate r, the reward for innovating needs to be discounted by the factor e−yr . A ﬁrm i chooses its R&D intensity Ii (ω, t) by maximizing the expected reward for innovating (since innovation success is not certain but occurs with instantaneous probability Ii ) minus costs at each point in time. Hence, it maximizes e−yr ϑIi − wH hi , with Ii = ηhi /X, through the optimal choice of hi . Due to free entry into the R&D sector, these expected proﬁts are competed away, and in equilibrium we must have the skilled wage rate deﬁned by the “R&D equilibrium condition” wH = e−yr

e−yr η c(1 − 1/λ) − IB ηϑ , = ˙ k X r+I − ϑ

(12)

ϑ

which is valid in any industry ω by symmetry of the model. 2.3 Job Matching and Unemployment The details of our model concerning matching and unskilled “Schumpeterian” unemployment are a slightly changed, closed-economy version of those in S¸ener (2001) who in turn builds on Aghion and Howitt (1994). Unskilled workers are continuously engaged in an onthe-job search, while ﬁrms can only start searching for the appropriate

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unskilled worker to use the new production technology after having innovated. Thus, they start producing and reaping monopoly proﬁts with an exogenously given delay of length y > 0. The matching process is deterministic, and the successful matching of technology leaders with unskilled workers follows a “well-behaved” matching function Am(Vt , Lt ), where A > 0 is a constant eﬃciency parameter.7 This matching function is concave, homogenous of degree one and increasing both in vacancies Vt created by innovating ﬁrms and the total unskilled workforce Lt . The aggregate job-ﬁnding rate (per eﬃciency unit A of the matching function) p(νt ) measures the rate at which a worker who starts search at any day ﬁnds a new job. This rate is deﬁned as p(νt ) = m(ν, 1) > 0 ∀ νt > 0 with the vacancy rate νt = Vt /Lt , and we assume that p(0) = 0 and p(∞) = ∞ holds.8 The aggregate recruiting-success rate (per eﬃciency unit A of the matching function) q(νt ) measures the rate at which a ﬁrm that has just innovated successfully matches a worker with the new technology. This rate is deﬁned as q(νt ) = m(νt , 1)/νt > 0 ∀ νt > 0, and we assume that q(∞) = 0 and q(0) = ∞ holds. q(νt ) and y are closely related as will be discussed in Sect. 3 below. The aggregate rate of change in the number of vacancies V within a time interval [t, t + dt] is given by ct+y Nt+y ct+y Nt+y βdt − Am(Vt , Lt )dt. It dt + V˙t dt = λ λ

(13)

The ﬁrst expression on the RHS of (13) gives the ﬂow of newly posted vacancies at time t equal to the expected unskilled labor demand at time t + y that occurs with the innovation probability It . This term follows from the production function that assumes a one-to-one relationship between unskilled labor input and consumer goods output, and the unit-elastic consumption demand. Furthermore, it takes into account that positions can only be ﬁlled after a matching time y, when total consumption has changed. The second expression on the RHS of (13) takes into account the change of the market size within the time interval [t+y, t+y +dt], that is, for the growing unskilled labor demand after production of the new leading-quality goods begins. In that interval, population grows at the exogenous rate n = β −δ > 0 (with β and δ denoting the birth rate and the death rate, respectively), and positions 7 In order to avoid the problem that workers may be matched while still being employed, we have to assume that (deterministic) matching time is long enough, as in Aghion and Howitt (1998, p. 126, fn. 7). 8 The only diﬀerence to S ¸ ener (2001) is that he considers the special case A = 1. Here, we introduce the matching-productivity parameter A in order to have an exogenous parameter that allows us to perform a comparative static exercise with respect to the eﬃciency of the matching process (in Sect. 5.1 below).

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of dying employed unskilled workers must be ﬁlled. The third expression on the RHS of (13) denotes outﬂow of vacancies due to successful matching during dt. Note that vacancy creation at time t is determined by additional labor demand at time t + y, but it is determined by the value of the matching function at time t. Similarly, the aggregate rate of change in the unemployment level U ≡ uL within a time interval [t, t + dt] is given by ct Nt It dt + βLt dt − δut Lt dt − Am(Vt , Lt )dt. U˙ t dt = λ

(14)

The four terms on the RHS of (14) denote, respectively, layoﬀs by the current quality leader in any industry ω in the case of innovation by a competing ﬁrm, newly born individuals who decide not to take education before they enter the labor market, dying unemployed workers, and outﬂow of unemployment due to successful matching, all measured during the time interval dt. Thus, the total unemployment level consists of Schumpeterian and demographic unemployment. Firing costs do not directly aﬀect vacancy creation or unemployment ﬂows, but they will have indirect eﬀects via the endogenous variables It , ct and ut .

3 Steady State and Rising Firing Costs We deﬁne the steady state of our model as the long-run equilibrium with market clearing and constant values for our variables θ0 , wH , I, c, v and u, given the level B = F/N of ﬁring costs to be determined later by the majority-winning preferences of workers. We begin by deriving the steady-state unemployment rate, following S ¸ ener (2001). Dividing (14) by dt, using m(V, L) = p(ν)L, L = θ0 N, U/L = u and (6) yields (1 − u)I + β − δu − Ap(ν) U˙ = β − δ = n, = u U

(15)

where the second equality follows from imposing the restriction of a constant steady-state unemployment rate. Equation (15) can easily be solved for u=1−

Ap(ν ∗ ) I +β

(16)

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with ν ∗ denoting the steady-state vacancy rate.9 This can be derived in a similar fashion from (13), using V˙ = β − δ, ct+y = ct = c, Nt+y = Nt eny V and the result in (16): ν∗ =

Ap(ν ∗ )(eny − 1) . n

(17)

The aggregate recruiting-success rate q and the matching time y are related as follows (cf. S¸ener, 2000, pp. 579-80): Aq(νt ) =

n Ap(νt ) Am(νt , 1) = ny = e −1 νt νt

(17a)

where the ﬁrst two equations follow from our deﬁnitions of q(ν) and p(ν), and the last equation follows from (17). Whereas Aq(νt ) is a monotonically decreasing function of νt , n/(eny − 1) is a positive constant, which deﬁnes uniquely the steady-state value ν ∗ as a function of exogenous parameters. In the limit n → 0, we obtain Aq(νt ) = 1/y by applying L’Hˆ opital’s rule. This is the special case (for A = 1) considered in Aghion and Howitt (1994): with no population growth, the matching time y is simply the inverse of the aggregate recruiting-success rate 1/q, which is the instantaneous expected matching time for a ﬁxed pool of searching workers. With n > 0, however, it holds y < 1/q since ﬁrms can choose from a continuously growing pool of searching workers (note that n/(eny − 1) decreases in n). In Sect. 5.1 we will conduct comparative statics on A with keeping ν ∗ constant, so that the value of y is adjusted accordingly. The only endogenous variable aﬀecting (positively) the unskilled unemployment rate u in (16) is the innovation rate I: more frequent innovations reduce the expected incumbency period 1/I of current quality leaders in any industry ω, thereby reducing the duration of employment spells of unskilled workers and raising their unemployment rate by increased labor market turnover.10 Note that the ﬁnding of 9

Similar to S ¸ ener (2001, p. 136), we note that the additional assumption Ap(ν) > β is needed to rule out that “biological turnover” alone suﬃces to produce a positive unemployment rate. 10 There are two reasons for why in addition to this positive “creative destruction eﬀect” of growth on unemployment, there is no negative “capitalization eﬀect”. First, as argued by Aghion and Howitt (1998, p. 127), since the quality leader does not upgrade its own technology, the capitalized value of each innovation does not grow with higher labor productivity (in terms of consumer goods quality), hence no additional incentives of follower ﬁrms to create vacancies arise. Second, unlike in the model of Aghion and

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an unambiguously positive long-run relationship between growth and unemployment hinges crucially on the fact that we are only considering the quality dimension of growth (increasing quality of consumer goods), while neglecting the quantity dimension. The latter could be introduced e.g. by accounting for human capital accumulation, with the level of human capital entering the production function of goods. Then, growth in the quantity dimension would raise production and hence the unskilled labor demand for any given innovation rate, thereby working toward a reduction in unemployment. Next, we will determine the steady-state values for both I and θ0 . To this end, we solve (12) for I, use (6) to replace c, (5) to replace wH , (16) to replace u, and use rt = ρ. Proceeding in this manner, we get I=

ηe−ρy Ap(ν ∗ )θ02 (λ − 1) − (ρ − n)σAp(ν ∗ )k . σAp(ν ∗ )k + (β + I)ηe−ρy θ0 B

(18)

In Appendix A we prove by use of the implicit function theorem that dI/dθ0 > 0 holds in (18). A second equation in the variables I and θ0 is derived by the skilled labor market equilibrium condition. Whereas skilled labor demand equals IkN/η from (7) and (8), skilled labor supply can be derived as (1−θ02 )φN/2 with φ ≡ [en(D−T ) −1]/(enD −1) < 1 (see Dinopoulos and Segerstrom, 1999, p. 456). Putting both together, the skilled labor market clearing condition is / 2kI . (19) θ0 = 1 − ηφ Equations (18) and (19) together determine the unique steady-state combination (I, θ0 ) as depicted in Fig. 1 (next page).11 There, we also illustrate the following (formal proof is provided in Appendix B): Proposition 1. A rise in ﬁring costs reduces unambiguously both the steady-state innovation rate and the proportion 1−θ0 of skilled workers. It will prove useful to derive the steady-state equilibrium also in (wH , I)-space. This is done in Fig. 2 (page 91). To derive Fig. 2, we Howitt (1994, p. 483; 1998, p. 135), the capitalization eﬀect does not enter when considering forward-looking behavior of ﬁrms. This is because in (10), the instantaneous expected income from innovating is growing with the size of the market (n) which does not depend on the rate of innovation. Therefore, a higher growth and innovation rate does not reduce (actually, it increases) the net discount rate at which quality leaders capitalize their expected proﬁts. 11 S ¸ ener (2001, p. 138, Fig. 1) studies the steady-state eﬀects of tradeliberalization using exactly the same graph.

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Wolf-Heimo Grieben q0 (18)

1 +D B

(18)

( r - n )× s × k h × (l - 1)× e - r × y (19) h ×f 2×k

I

Fig. 1. Steady-state eﬀects of rising ﬁring costs in (I, θ0 )-space

ﬁrst solve (12) for I, use (6) to substitute for c, (5) to substitute for 1 − u, and (19) to substitute for θ0 . This yields η −ρy e (λ − 1) − (ρ − n)k , I = σ 2e−ρy (λ−1) ηe−ρy B + k 1+ wH σφ

(20)

which is unambiguously decreasing in B. Then, we use (5), (16) and (19) to write wH as a function of I alone: wH =

σAp(ν ∗ ) . 1/2 (β + I)(1 − 2kI ηφ )

(21)

In (21), wH is decreasing in I for I < (θ02 ηφ/k) − β and rising for larger I. Equations (20) and (21) together determine the unique steady-state combination (I, wH ) as depicted in Fig. 2. Equation (20) captures the R&D equilibrium with skilled labor market clearing. Its positive slope indicates that the higher the innovation rate in equilibrium, the higher the skilled wage rate will be due to the rising demand for skilled workers. Equation (21) accounts for the equilibrium education decision of individuals, the steady-state unemployment rate and the skilled labor market clearing condition. It gives for each level of the equilibrium innovation rate (that implies an equilibrium unemployment rate via (16)) the required skilled wage rate that makes the individual with threshold ability θ0 indiﬀerent with respect

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b

(20)

s × A× p(v* )

(20)

wH

(2 1

)

+D B

2

I crit =

q 0 ×h × f -b k

I

Fig. 2. Steady-state eﬀects of rising ﬁring costs in (I, wH )-space

to his educational decision, given skilled labor market clearing. This curve is U -shaped. For low levels of I, on the one hand, unskilled unemployment is relatively low, hence expected unskilled wage income is relatively high, which raises the required skilled wage rate (“education equilibrium eﬀect”). On the other hand, skilled labor market clearing would call for a low level of wH (“skilled labor demand eﬀect”) because demand for R&D labor is low. For low I, the education equilibrium eﬀect dominates the skilled labor demand eﬀect, and this is why the required wH is high in this case. For high levels of I, unskilled unemployment is high, thus expected unskilled wage income is low. Ceteris paribus, this yields a low required wH . However, this education equilibrium eﬀect is dominated by the skilled labor demand eﬀect: a high I creates a large demand for skilled workers that drives up wH . This is why the required wH is high for large I. For I = (θ02 ηφ/k) − β, both eﬀects together imply the minimum level of wH consistent with (5), (16) and (19). In addition to this, Fig. 2 also illustrates the following: Proposition 2. An increase in ﬁring costs raises the skilled wage rate for I < I crit ≡ (θ02 ηφ/k) − β and reduces it for I > I crit . Proposition 2 simply follows from the fact that an increase in ﬁring costs raises unambiguously the denominator of (20), thus this curve shifts to the left in Fig. 2 without aﬀecting the curve for (21). As drawn,

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the initial steady-state innovation rate is below the critical value I crit = (θ02 ηφ/k) − β. To see whether this assumption is plausible, we assign the following benchmark simulation values to our parameters, taken from Dinopoulos and Segerstrom (1999, p. 467): a population growth rate of n = 0.01, a time preference rate ρ = 0.03 to get a 3-percent steady-state real interest rate r = ρ, a working life of D = 40 years12 , a duration of college education of T = 4 years13 (implying σ = 1.193 and φ = 0.881), and a 35-percent improvement in product quality through each innovation (λ = 1.35). As noted by Steger (2003, p. 4), average estimates about the monopolists’ percentage markup over marginal costs λ − 1 lie in the interval [0.1, 0.4], hence reasonable values for λ are in the range [1.1, 1.4]. In addition, we also follow Dinopoulos and Segerstrom (1999) by imposing a 2-percent steady-state utility growth rate14 z/z ˙ = I log λ = 0.02, from which we get I = 0.0666. It remains to derive a parameter value for η/k and a value for θ0 . The latter is determined endogenously by (18) and (19). Even for a given I = 0.0666, this yields θ0 as a function of y, A, p(ν ∗ ) - all of which are unknown and not straightforward to derive - and of B which will be derived endogenously in Sect. 4 below. Therefore, we prefer to choose θ0 from the data in order to derive a ‘plausible’ value for η/k.15 The OECD (2003, Table A2.4) reports trends in educational attainment at the tertiary level from 1991 to 2001, which corresponds to 1−θ0 in our model. We take the most recent values from 2001 for the US, Germany and Italy, where the percentage of the population of 25 to 34-year-old males that have attained tertiary education equaled 36, 23 and 10, respectively. This implies that the threshold ability level 12

This yields a birth rate of β = nenD /(enD − 1) ≈ 0.03. See Dinopoulos and Segerstrom (1999 p. 454) for a derivation. 13 Hence, a “skilled worker” is deﬁned as having taken four years of college, whereas unskilled workers only have a high-school degree. 14 As is usual in quality-ladder growth models, this steady-state growth rate is found by noting R s that the expected number of innovations up to time t equals R1 j(w, s)dω = 0 I(t)dt. Using this in the instantaneous utility function (2) 0 and diﬀerentiating with respect to time gives the result. 15 Hence, our procedure is explained by help of Fig. 1 as follows: ﬁrst, we impose a ﬁxed value for the innovation rate I that follows from the assumption of a two percent steady-state utility growth rate, depending on a particular value for λ. This means that we draw a vertical line for a speciﬁc I instead of the curve for (18). The intersection of this vertical line with the curve for (19) deﬁnes θ0 , but this depends on the value for η/k. Second, we impose a value for θ0 as suggested by the data. Given I and θ0 , a “plausible” value for η/k follows from (19), which deﬁnes the position of this curve in Fig. 1. Note that our only purpose of assigning speciﬁc values to θ0 from the data is to derive values for η/k. In Sect. 4 below, we will use a speciﬁc value for η/k and derive both I and θ0 endogenously.

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for the US (Germany, Italy) is θ0 = 0.64 (0.77, 0.90).16 Given this, a value for η/k follows for given I from (7) and (8): I = η(1 − θ02 )φ/(2k). Table 1 (page 94) provides a robustness check for the comparison of I and I crit .17 Table 1 indicates for each λ the critical threshold ability level θ0crit for which I = I crit holds. The table shows that our implicit assumption I < I crit in Fig. 2 does (not) hold in the case of the US for low (high) values of λ (the bold numbers indicate a violation of this condition; for λ = 1.2912, I = I crit holds for given θ0 = 0.64). For the cases of Germany (θ0 = 0.77) or Italy (θ0 = 0.90), the condition I < I crit is fulﬁlled over the whole range λ ∈ [1.1, 1.4]. Thus, in countries with a relatively high proportion of people who have attained tertiary education (e.g. Japan, Canada), an increase in ﬁring costs reduces the skilled wage rate, i.e., the skilled labor demand eﬀect dominates the education equilibrium eﬀect. This implies that the skilled wage rate and the innovation rate move in the same direction after a change in ﬁring costs. In countries with a relatively low proportion of people who have attained tertiary education (e.g. France, UK, Germany), the skilled wage rate and the innovation rate move in opposite direction after a change in ﬁring costs. In all cases, after an increase in ﬁring costs, the declining innovation rate implies a decrease in the unskilled unemployment rate because of slower labor market turnover (see (16)), which ceteris paribus induces fewer individuals than before to invest in education since the expected unskilled wage income increases. In the case of I < I crit (I > I crit ), the rise (decline) in the skilled wage rate works in the opposite (same) direction. In any case, the net eﬀect on θ0 is positive as depicted in Fig. 1. However, the net eﬀect on the aggregate unemployment rate ua = θ0 u in principle is ambiguous.18 Finally, the decline in unskilled unemployment and the rise in the proportion of unskilled workers both increase 16

From Table A2.4 in OECD (2003), we derive the following threshold ability levels of other major OECD countries for males in 2001: θ0 = 0.54 in Japan, θ0 = 0.55 in Canada, θ0 = 0.66 in Sweden, θ0 = 0.68 in France, θ0 = 0.70 in the UK, and θ0 = 0.74 in the Netherlands. In 1991, the lowest values for males available in the OECD dataset were θ0 = 0.70 for Canada and θ0 = 0.71 for the US. 17 Note from Fig. 1 that the maximum feasible innovation rate consistent with an interior steady-state equilibrium is below I max = ηφ/(2k). In Table 1, I crit > I max holds in the case of Italy (θ0 = 0.90) for the whole range λ ∈ [1.1, 1.4], and in the case of Germany (θ0 = 0.77) for all λ except λ = 1.4. 18 Using (19) and (16), with I as given in (18) and dI/dB < 0 from Proposition 1, it is straightforward to derive the following condition for a rise in ﬁring 1 (1 − u) − βu]. [ ηφ costs to reduce the aggregate unemployment rate: I < 2−u k Given our benchmark values (implying I = 0.0666) and the intermediate

94

Wolf-Heimo Grieben Table 1. Robustness check for the comparison of I and I crit

λ = 1.4: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6559 λ = 1.35: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6490 λ = 1.3: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6414 λ = 1.25: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6332 λ = 1.2: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6242 λ = 1.15: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6143 λ = 1.10: θ0 = 0.64 θ0 = 0.77 θ0 = 0.90 θ0crit = 0.6033

I 0.0594

η/k

I crit

I max

0.2285 0.3314 0.7101 0.2368

0.0521 0.1428 0.4765 0.0594

0.1007 0.1460 0.3128 0.1043

0.2562 0.3716 0.7962 0.2614

0.0621 0.1638 0.5379 0.0666

0.1129 0.1637 0.3508 0.1152

0.2931 0.4251 0.9107 0.2940

0.0754 0.1917 0.6196 0.0762

0.1291 0.1873 0.4012 0.1295

0.3446 0.4998 1.0708 0.3396

0.0940 0.2307 0.7339 0.0896

0.1518 0.2202 0.4717 0.1496

0.4218 0.6117 1.3106 0.4080

0.1219 0.2892 0.9050 0.1097

0.1858 0.2695 0.5774 0.1797

0.5502 0.7979 1.7096 0.5217

0.1682 0.3865 1.1898 0.1431

0.2424 0.3515 0.7531 0.2298

0.8068 1.1701 2.5070 0.7489

0.2608 0.5809 1.7588 0.2098

0.3554 0.5154 1.1044 0.3299

0.0666

0.0762

0.0896

0.1097

0.1431

0.2098

eﬀective unskilled labor supply, which increases goods production and thus consumption quantity q = c/λ, see (6).

case of Germany with η/k = 0.3716 (derived from setting θ0 = 0.77), this condition is fulﬁlled for unskilled unemployment rates up to u < 0.6676.

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4 Preferences Regarding Firing Costs In this Section we derive the preferences regarding ﬁring costs of skilled and unskilled workers in a once-and-for-all vote on B = F/N , while taking into account that with forward-looking agents, all individuals decide concurrently about their education and their vote on B. This requires three steps of procedure: First, we consider the education decision of a household by comparing the two corresponding discounted lifetime utility streams. This comparison reveals that all households with ability θi < θ0 (I) (θi > θ0 (I)) prefer to stay unskilled (to become skilled). Second, we maximize separately the discounted (expected) lifetime utility of a skilled and an unskilled household with respect to B under the assumption of a given skill choice. Third, we show that those households who lose the ballot under majority voting (i.e., the skilled) can do no better than staying with their educational decision, even if they perfectly foresee that they would lose the ballot. The level of per-capita ﬁring costs demanded by the majority of unskilled households will be implemented, which results unambiguously in a particular “desired” innovation rate. Depending on this innovation rate, the threshold ability level θ0 (I) adjusts according to (19). We will restrict the analysis to a steady-state welfare comparison. This somewhat limits the scope of our analysis, because when assuming a particular starting value B0 of per-capita ﬁring costs, the welfare comparison should take the transitional period into account.19 At the end of this section we argue that the implied imprecision is unlikely to change our results qualitatively. Since all individual members of a given household are identical, maximizing discounted lifetime utility of a representative household member is the same as maximizing it for the whole household. Since ﬁnitely living household members behave like 19 Since the transitional dynamics are analytically intractable in this kind of model, one way is to rely on numerical simulations similar to, e.g., Joseph and Weitzenblum (2003). In a model of unemployment, moral hazard and precautionary savings, they simulate the transitional welfare eﬀects of a cut of the replacement rate, ﬁnanced by a tax and paid to unemployed workers. Such an analysis is beyond the scope of our paper. Another way out is to focus on “stationary political equilibria” as suggested by Saint-Paul (2002, pp. 687-88). This type of equilibrium has two properties: ﬁrst, we start in a (unique) steady-state equilibrium for a given level of per-capita ﬁring costs B. Second, this initial level of per-capita ﬁring costs equals the majoritywinning level, which in our case will be the level preferred by all individuals who choose to stay unskilled under these ﬁring costs. Starting from there, a once-and-for-all vote on the level of B would aﬃrm the current equilibrium. Saint-Paul considers this

“[...] as the limit steady state of a repeated voting equilibrium as the frequency of voting goes to zero” (ibid, p. 687). In Sect. 6.4 we will argue that in our particular case, the frequency of voting is irrelevant for the steady-state level of ﬁring costs that we obtain.

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a dynastic family, this maximization is done over an inﬁnite horizon. Therefore, the particular age of the (unskilled) median voter does not matter. Due to the assumption of perfect unemployment insurance, the employment status of an unskilled voter also does not matter. Let us consider ﬁrst the voting problem of a household in case its members decide to stay unskilled under a given level of ﬁring costs that results from majority voting. Inserting instantaneous utility (2) into total discounted utility (1), and using p = λ and

1

s

j(w, s)dω = 0

I(t)dt 0

yields Z(θi < θ0 ) = [1/(ρ − n)][log(1 − u) − log λ + I log λ/(ρ − n)]

(22)

as discounted lifetime utility of an unskilled household with ability θi < θ0 for all of its members. Since workers do not own ﬁrms, the expected consumption expenditure per period of this household is simply (1 − u)wL = 1 − u. Using this and (16) in (22) gives:

I(B) log λ * Ap(ν ∗ ) 1 ) . (23) − log λ + log Z(θi < θ0 ) = ρ−n β + I(B) ρ−n Diﬀerentiating (23) with respect to B yields the f.o.c.

dI log λ * Ap(ν ∗ )dI/dB 1 ) β + I(B) dZ(θi < θ0 ) = 0, + − = dB ρ − n [β + I(B)]2 ρ − n Ap(ν ∗ ) dB from which we derive the aggregate innovation rate desired by all unskilled workers: ˆ i < θ0 ) = [(ρ − n)/ log λ] − β. I(θ

(24)

This is the innovation rate that optimally balances the unskilled workers’ conﬂicting interests in rising consumption quantity versus rising growth of consumer goods quality. Any slightly faster rate of technical progress would result in a net utility loss because the marginal utility gain of faster quality growth (increase in the third term in curly brackets in (23)) would be more than oﬀset by the marginal utility loss of lower consumption quantity (decrease in the ﬁrst term in curly brackets in (23)). The argument for the case of a slightly lower rate of technical change is analogous. Using our benchmark parameter values, (24) imˆ i < θ0 ) = 0.0363 which is well below the innovation rate used plies I(θ above I = 0.0666 to generate a 2-percent steady-state utility growth

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rate. Inserting (24) into (18) and (19) allows us to solve the model for the unique per-capita ﬁring costs preferred by all unskilled workers: ˆ i < θ0 ) = Ap(ν ∗ ) B(θ

ρ−n ˆ i < θ0 )] ηθ0 [I(θ log λ

ρ−n −β log λ

−1

ˆ i < θ0 )]2 − σkeρy [(ρ − n)(1 + 1/ log λ) − β]}, (25) {η(λ − 1)θ0 [I(θ ˆ i < θ0 )] denotes the threshold ability level that results afwhere θ0 [I(θ ter inserting (24) in (19).20 If θ0 > 0.5 holds, (25) would also give the level of per-capita ﬁring costs preferred by the majority of all workers.21 Given equal participation rates of skilled and unskilled workers in a vote on B, the median voter will be unskilled, hence (25) will be ˆ i < θ0 ) is rising in realized. In Sect. 5.1 we will exploit the fact that B(θ the matching eﬃciency parameter A to analyze eﬀects of a change in the exogenous (A) on the endogenous (B) component of labor market ﬂexibility. Let us now consider the voting problem of a household in case its members decide to become skilled under a given level of ﬁring costs that results from majority voting. The discounted lifetime utility of a skilled household with ability θi ≥ θ0 of all its members is Z(θi ≥ θ0 ) = [1/(ρ − n)][log(θi wH ) − log λ + I log λ/(ρ − n)]. 20

(26)

For our benchmark parameter values and η/k = 0.3716 from the intermeˆ i < θ0 )] = 0.8821. Then (25) implies diate case of Germany, we derive θ0 [I(θ ˆ i < θ0 )/∂A > 0 which will be used a positive value of ﬁring costs (and ∂ B(θ later) for a matching time up to y ≤ 13.4 years. Finally, in order to check whether the inequality (11) is fulﬁlled, we have to assign additional speciﬁc parameter values for y, p(ν ∗ ) and A. A value for p(ν ∗ ) is imposed by assigning an empirically ‘plausible’ value for the unskilled unemployment rate in (16). According to Reinberg and Hummel (2003, Table 1b), the average unemployment rate among West German males aged 15-64 with no ﬁnished vocational education has been 20.1% within the period 1987 to 2002. Given ˆ i < θ0 ) = 0.0363 and imposing an unskilled unemployment rate u = 0.2 I(θ and A = 1, (16) yields p(ν ∗ ) = 0.0533. Using this and a matching time as ˆ i < θ0 ) is decreasing in y for all λ ∈ [1.1, 1.4]), low as y = 0.1 (note that B(θ ˆ i < θ0 )I(θ ˆ i < θ0 ) ≈ 0.0825 < c(λ − 1)/λ = we verify by using (6) that B(θ (λ − 1)(1 − u)θ0 ≈ 0.2467. 21 Since θ0 is endogenous, we need to state the assumption θ0 > 0.5 in terms of the exogenous parameters. Inserting (24) into (19), this assumption log λ]−β . For the whole range of parameter values is equivalent to 38 > [(ρ−n)/ (η/k)φ for λ and η/k that is covered by Table 1, this assumption is clearly satisﬁed (given our benchmark parameter values for ρ, n, β and φ). Moreover, the empirical evidence given in OECD (2003, Table A2.4) suggests threshold ability levels between θ0 = 0.54 (Japan) and 0.90 (Italy) for males in 2001.

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The expected consumption expenditure per period of this household is simply θi wH . Using this and (21) for the skilled wage rate in (26) gives Z(θi ≥ θ0 ) =

I(B) log λ θi Ap(ν ∗ ) 1 0 . − log λ + log ρ−n ρ−n [β + I(B)] 1 − 2kI(B)/(ηφ) (27) The comparison of (27) and (23) immediately reveals that only those households with ability θi > θ0 (I) have the incentive to invest in education in order to realize Z(θi ≥ θ0 ) > Z(θi < θ0 ), ˆ i < θ0 ) under majority voting where θ0 (I) is given by (19), with I = I(θ for given θ0 > 0.5. Actually, this means that the minority of skilled ˆ i < θ0 )] as given. Finally, ˆ i < θ0 ) and thus θ0 [I(θ workers has to take I(θ i ˆ even when I = I(θ < θ0 ) is perfectly foreseen by all skilled households, they still prefer to become skilled and lose the vote on B rather than working as unskilled and joining the winning majority. Diﬀerentiation of (27) with respect to B yields the f.o.c. 1 dZ(θi ≥ θ0 ) = ρ−n dB

− θ0 dI dI log λ = 0, + dB ρ − n (β + I)θ0 dB

(β+I)k θ0 ηφ

(28)

where (19) and (21) have been used. We see that the individual characteristic θi cancels out, hence all skilled workers have identical preferences with regard to B. An optimum satisfying this f.o.c. only exists if the ﬁrst term in square brackets is negative (since dI/dB < 0), which is fulﬁlled if and only if I < I crit holds. From (28) we can derive the aggregate innovation rate desired by all skilled workers in implicit form:22 ˆ i ≥ θ0 ) = I(θ

(ρ − n)(ηφ − 2kI) −β (log λ)(ηφ − 2kI) + (ρ − n)k

(29)

with I < ηφ/(2k) from Fig. 1. Comparison of (29) with (24) gives our ﬁrst main result: 22 Using the benchmark parameters with the intermediate case of Germany (η/k = 0.3716), (29) can be transformed into the quadratic equation I 2 − ˆ i ≥ 0.233337I + 0.004934 = 0. This has two solutions: the larger one I(θ max θ0 ) = 0.2098 > I = 0.1637 is not feasible, whereas the smaller one ˆ i ≥ θ0 ) = 0.0235 < I(θ ˆ i < θ0 ) = 0.0363 < I max is feasible. I(θ

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Proposition 3. For I < I crit , skilled workers (i.e., those individuals who decide to become skilled after a majority decision on the level of ﬁring costs is reached), although not threatened by unemployment risk, (i) prefer unambiguously a lower steady-state innovation rate, and (ii) vote for a higher level of ﬁring costs than unskilled workers. For I ≥ I crit , skilled workers vote for zero ﬁring costs. Given part (i) of Proposition 3, part (ii) follows immediately from dI/dB < 0 as stated in Proposition 1. The intuitive reasoning for the surprising ﬁnding in part (i) of Proposition 3 is as follows. By voting on the level of ﬁring costs, both worker groups face a trade-oﬀ between rising consumption quantity and a slower growth rate of consumer goods quality. On the one hand, larger ﬁring costs reduce the aggregate innovation rate - the growth rate of the quality of consumer goods slows down, which reduces the discounted lifetime utility of both worker groups by the same amount (third term in (23) and (27)). On the other hand, larger ﬁring costs increase labor input in goods production by reducing the unskilled unemployment rate u and increasing the proportion θ0 of unskilled workers available. Each unskilled worker individually beneﬁts from the decline in u through the increase in expected wage income wL (1 − u) for the given wage rate wL ≡ 1. This eﬀect also beneﬁts skilled workers: a lower unskilled unemployment rate implies higher per-capita consumption expenditure c and thus a larger market for the output of the current quality leader, which raises the reward for innovating ϑ and thus wH . However, as argued when discussing Fig. 2, an increase in ﬁring costs triggers an education equilibrium eﬀect and a skilled labor demand eﬀect, and the former dominates the latter if and only if I < I crit holds. Thus, demand for skilled workers declines due to the lower innovation rate, but this is more than oﬀset by the required increase in wH in order to make the individual with ability θi = θ0 (with θ0 now being higher than before) indiﬀerent with respect to his educational choice. Hence, skilled workers beneﬁt from an additional gain through stricter employment protection relative to the unskilled. For I > I crit , however, both terms in the square brackets in (28) are negative, hence no interior solution exists. Since in this case a decrease in ﬁring costs raises both the growth rate of the quality of consumer goods and the skilled wage rate, skilled workers would vote for zero ﬁring costs (corner solution). Since skilled workers are still in the minority, the importance of Proposition 3 is not that it explains the observed cross-country variation in ﬁring costs, depending on the proportion of skilled workers

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in each country.23 However, the result may nevertheless be relevant in the following sense: as long as I < I crit holds, it oﬀers an argument for why we should not expect that the massive increase in the percentage of skilled workers that has been observed in most OECD countries over the last decades, induces political pressure to reduce employment protection levels for less-skilled workers. However, continuous skill upgrading reduces I crit (as shown in Table 1 above), and once I ≥ I crit holds, the (relatively large) proportion of skilled workers would vote or lobby for zero ﬁring costs. At the beginning of this section we argued that in principle, a complete welfare comparison should take into account the transitional peˆ i < θ0 ), we are riod. However, at least when starting from B0 < B(θ conﬁdent that taking account for transitional eﬀects would not change our results qualitatively for three reasons. First, a rise in per-capita ﬁring costs (e.g., from B0 to the level preferred by the unskilled workers) implies an increase in θ0 (see Fig. 1). Since fewer individuals decide to become educated, the education period of length T does not matter for the transition in this case. The required drop in the proportion of skilled workers H/L is achieved by the death of skilled workers, by the birth of unskilled workers who enter immediately the labor market, and by the immediate drop in the innovation rate, which results in declining unskilled unemployment. Second, the immediate decline in the unskilled unemployed rate raises production and consumption quantity of both skilled and unskilled workers. This positive welfare eﬀect dominates in the short run, whereas the negative welfare eﬀect (lower quality growth) arises mainly in the long run. Hence, in this case we do not face the problem that a negative short-run welfare eﬀect more than outweighs possible long-run welfare gains. In the opposite case of ˆ i < θ0 ), this problem arises and could prevent the majority B0 > B(θ of workers from voting in favor of a reform proposal aimed at reducing ﬁring costs, similar to the failure of the proposal to cut the replacement rate in the model of Joseph and Weitzenblum (2003) that we mentioned earlier. Third, the short-run eﬀect on consumption quantities works in the same direction for skilled and unskilled workers. Hence, there is no reason to believe that due to transitional eﬀects, our result in Proposition 3 should not continue to hold.

23 This conclusion may not hold anymore when considering the diﬀerent rates of inﬂuence held by skilled and unskilled workers in the political process. For instance, voter turnout is usually found to increase with income and, in particular, with education (Wolﬁnger and Rosenstone, 1980). Moreover, costly lobbying (campaign contributions) may rise with income which is positively related to skills.

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5 Determinants of the Level of Firing Costs 5.1 Interactions between Diﬀerent Types of Labor Market Rigidities When discussing possible reasons for the persistently higher unemployment rates in many European countries compared to the US, Siebert (1997) stressed the importance of taking account of interactions between diﬀerent types of labor market rigidities, instead of performing a ceteris-paribus analysis for each friction only: “Looking at single institutional characteristics one at a time is never very promising, because the eﬀect of a single institutional arrangement can only be understood in its interaction with other institutional rules. [...] Consequently, the cumulative eﬀect of rules is relevant determining their total impact.” (ibid, p. 39). We therefore analyze the eﬀects of a more eﬃcient matching technology from the viewpoint of both employers and employees, modeled as an increase in the exogenous matching productivity parameter A, on the steady-state equilibrium before and after a response of ˆ i < θ0 ), respectively. As is obvious from (17a), for given ν ∗ an inB(θ crease in A implies the simultaneous reduction of the matching time y. As shown in Fig. 3 (below), for a given level of per-capita ﬁring costs (18) q0

1

+ DA

-Dy

E0

E1

E2

( r - n )× s × k h × (l - 1)× e - r × y

(19) h ×f 2×k

I

Fig. 3. Steady-state eﬀects of more eﬃcient matching for ﬁxed B in (I, θ0 )-space

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B, an increase in A ﬁrst rotates the curve for (18) clockwise downward. Formally, this follows immediately from (18) when dividing both nominator and denominator by A. Then the resulting decrease in y shifts the entire curve for (18) further downward, including the ordinate intercept (proof for dI/dy < 0 is given in Appendix C). The innovation rate increases unambiguously, which implies a decline in θ0 , since more skilled workers are needed for R&D. The initial positive impact of a rise in A on the innovation rate comes from the reduction in unskilled unemployment, thus u determined in (16) decreases for a given level of I. Lower unemployment implies higher output and thus a larger market for innovating ﬁrms. More eﬃcient matching leads to a decrease in the matching time y. This also aﬀects positively the innovation rate because it implies a smaller discount rate on the innovators’ future monopoly proﬁts (production can start earlier) at any given value of the reward for innovating ϑ. The net eﬀect on unskilled unemployment is ambiguous since the increase in the innovation rate tends to raise u. The points {E0 , E1 , E2 } in Fig. 4 (below) correspond to those in Fig. 3. Initially, the skilled wage rate increases with rising A due to the drop in unskilled unemployment that raises the proﬁtability of doing R&D (move from E0 to E1 ). The resulting decrease in matching time y triggers an education equilibrium eﬀect and a skilled labor demand

(20)

wH

s × A × p (v* ) b

-Dy

E1 E0

) ( 21

E2 +DA

I

Fig. 4. Steady-state eﬀects of more eﬃcient matching for ﬁxed B in (I, wH )-space

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eﬀect, and the former dominates the latter for I < I crit . Thus, demand for skilled workers rises due to the larger innovation rate, but this positive eﬀect on wH is more than oﬀset by the required decrease in wH in order to make the individual with ability θi = θ0 (with θ0 now being lower than before) indiﬀerent with respect to his educational choice. The required decrease in wH comes from the fact that an increase in I raises u which reduces wL (1 − u). Hence, a lower wH is needed to induce all individuals with θi > θ0 to become educated (move from E1 to E2 in Fig. 4). The net eﬀect on wH is ambiguous for I < I crit , and unambiguously positive for I > I crit . Since the matching time is reduced, the increase in A can be interpreted as an increase in labor market ﬂexibility. This raises unambiguously the level of ﬁring costs preferred by the unskilled workers ˆ i < θ0 )/∂A > 0 follows from who form the majority. Formally, ∂ B(θ ˆ i < θ0 ) > 0 holds. This could be illustrated by help (25) whenever B(θ ˆ i < θ0 ) holds at the initial of Fig. 3 as follows: suppose that I = I(θ steady-state equilibrium E0 . After the (unexpected) change in A and y, unskilled workers (i.e., those who decide not to invest in education ˆ i < θ0 ) such that the under the new B) vote for a new value for B(θ curve for (18) rotates counterclockwise around its ordinate intercept ˆ i < θ0 ) is reestablished at E0 again. Hence our second until I = I(θ main result follows: Proposition 4. An exogenous increase in labor market ﬂexibility, ceteris paribus accelerating innovation and economic growth, increases the political support for larger ﬁring costs of unskilled workers (i.e., of those individuals who decide not to become skilled after a majority decision on the level of ﬁring costs is reached for the given new value of A), thereby reducing innovation and growth again. Hence, a more ﬂexible labor market with more eﬃcient matching has two eﬀects: ﬁrst, it induces an increase in the aggregate innovation and growth rate for a given level of ﬁring costs. Second, it increases the level of per-capita ﬁring costs preferred by the majority-winning unskilled workers, which in turn reduces the steady-state innovation and growth rate.24 The intuition for this result is as follows: for the optimal decision on B, the net eﬀect of an increase in matching eﬃciency A on the unskilled workers’ unemployment rate (which is ambiguous for given B) does not matter. The welfare-maximizing choice of per-capita 24

Suppose that I < I crit holds. Then, skilled workers (i.e., those individuals ˆ i < θ0 )) vote for who would still decide to become skilled under the new B(θ a level of ﬁring costs that exceeds the one preferred by the unskilled by a constant ‘markup’ as is obvious from comparing (29) and (24). Hence they also increase their demand for ﬁring costs after a rise in A. The intuition is similar to that given for the unskilled workers.

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ﬁring costs B results in the unique desired innovation rate (24) that is independent of A and y - the marginal eﬀects of a change in B on the lifetime utility derived from consumption quantity and quality growth do not change. Since the increase in A raises the steady-state innovation rate, optimal behavior of the unskilled (as of the skilled) workers is to restore the welfare-maximizing path of I, which requires that ﬁring costs increase accordingly. A positive eﬀect on the level of discounted ˆ i < θ0 ) is relifetime utility of all workers will remain after I = I(θ stored again. This is because it remains the increase in A in the ﬁrst term in curly brackets of (23) and (27), respectively, and with ﬁxed ˆ i < θ0 ), an increase in A reduces u unambiguously. I = I(θ ˆ i < θ0 ), an increase Hence, given that the economy starts at I = I(θ in matching eﬃciency would not change the steady-state innovation and growth rate at all if a once-and-for-all vote on the level of B took place ˆ i < θ0 ) is preserved thereafter. In Sect. 6.4 we will argue that I = I(θ even with repeated voting. The result presented in this section serves as an example for the interaction of labor market frictions as stressed by Siebert (1997) cited above. Here, one rigidity of the labor market is the non-instantaneous matching of unemployed workers with producing ﬁrms as deﬁned by the matching function Am(V, L). More eﬃcient matching works toward an alleviation of this rigidity. However, this intensiﬁes a second labor market rigidity, namely the level of requested ﬁring costs: alleviating one (exogenous) rigidity aggravates the other (endogenous) one.

5.2 The Trade-oﬀ between Growth and Employment Protection In his vintage model of exogenous growth, Saint-Paul (2002) ﬁnds that the workers’ gains from employment protection are smaller the higher the growth rate, i.e., the faster the rate of creative destruction.25 The reasoning for his ﬁnding is as follows. Employment protection keeps a certain fraction of the workforce in vintages with productivity below that of the most recent vintage, because in his model, ﬁrms determine endogenously the optimal exit date which is postponed by larger ﬁring 25 This may be viewed as a contrast to his own empirical ﬁndings (SaintPaul, 1996, pp. 283-86). There, he ﬁnds the empirical regularity present in EU countries that, since the 1960s, governments reduced (increased) ﬁring costs at times of low (high) growth. Given that the government did follow the workers’ demand for employment protection, two explanations for this apparent contradiction seem plausible. Either the sign of the (theoretical) correlation between growth and requested ﬁring costs should be positive as is found in Saint-Paul’s data, or his empirical ﬁndings reﬂect only shortsighted motivation of workers when voting on ﬁring costs. Our model takes a long-run perspective.

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costs.26 Faster growth enlarges the productivity gap between old and new vintages, whereas the opportunity costs of working (that is, the value of being unemployed) are increasing at the current growth rate. Hence, the value of keeping the current employment (and reaping the rent that arises by assumption of match-speciﬁc human capital) deteriorates faster with higher economic growth, which reduces support for employment protection by the decisive median voter. Since the growth rate is endogenous in our model, the comparative static exercise is less straightforward than in Saint-Paul’s model. However, to analyze workers’ attitudes toward employment protection in a faster growing economy, we can look at the eﬀects of changes to the exogenous parameters closely related to growth, namely a rise in the size of innovations λ and a rise in the R&D productivity parameter η. The marginal eﬀect of a larger size of innovations on the steady-state utility growth rate z/z ˙ = I log λ is positive for ﬁxed B: I1 ∂(z/z) ˙ ∂I1 (log λ) + = λ ∂λ ∂λ −1/2

2 I1 Ap(ν ∗ ) P 1 > 0, θ0 (log λ) + −Q = λ B 2 2 where I1 ≡ −P/2 +

0

(30)

((P/2)2 − Q) > 0

1 with P > 0 and Q < 0 as deﬁned in Appendix B. Due to ∂I ∂λ > 0, an increase in λ shifts the curve for (18) downward in Fig. 1 without aﬀecting the curve for (19). Hence, for given B, a larger size of innovations speeds up innovation and growth. However, diﬀerentiation of ˆ i < θ0 )/∂λ > 0 holds.27 Therefore, the results (25) shows that ∂ B(θ

26 More precisely, the optimal exit date is determined by equalizing the marginal loss per unit of time (wage minus productivity) to the annuity equivalent of the ﬁring costs. Thus, in Saint-Paul’s model, production takes place with several technologies simultaneously, and technical progress improves the eﬃciency of the production process without aﬀecting the quality of consumer goods. Therefore, the consumption side of the economy can be (and actually is) neglected in his framework. In our model, by contrast (that explains ‘creative destruction’ endogenously), technical progress involves product innovations that result in rising consumer goods quality over time. Since we assume limit pricing behavior of the quality leader, and consumers only buy goods with the lowest quality-adjusted price, consumer goods of diﬀerent quality levels do not coexist in our framework. Hence, rising ﬁring costs do not extend the lifetime of old ﬁrms but only the expected incumbency period 1/I of the current quality leader. 27 Whereas the nominator of (25) increases unambiguously with rising λ, a suﬃcient condition for that the denominator of (25) declines with rising λ is 1 ≥ [k/(ηφ)]{3[(ρ − n)/ log λ] − 2β}. For our benchmark parameter values, the RHS of this inequality takes a value of 0.4254. Moreover, this inequality is satisﬁed for any possible values for λ and η/k considered in Table 1.

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of an increase in λ are similar to that of an increase in the matching eﬃciency (Proposition 4). The only diﬀerence in this case is that the growth rate desired by unskilled workers (i.e., by those individuals ˆ i < θ0 )) is aﬀected: who do not invest in education under the new B(θ i 2 ˆ ∂ I(θ < θ0 )/∂λ = −(ρ − n)/[(log λ) λ] < 0. If each innovation raises product quality to a larger extent, workers would adjust their vote on ﬁring costs as to reduce the desired expected frequency of innovations. This is easily understood by looking at (23): on the one hand, a larger size of innovations raises the marginal utility gain of a slightly increasing innovation rate in terms of higher quality growth (third term in curly brackets of (23)). On the other hand, the marginal utility loss of a rise in I in terms of declining consumption quantity is unaﬀected (ﬁrst term in curly brackets of (23)). Therefore, equality between marginal gains and losses is reached through a lower “desired” innovation rate, which requires to vote for a higher B.28 Hence the steady-state utility growth rate z/z ˙ = I log λ with voting on ﬁring costs declines: β ∂(ρ − n − β log λ) ∂(z/z) ˙ = − < 0, = | ˆ λ ∂λ ∂λ I=I(θ<θ0 )

(31)

ˆ i < θ0 )] increases according to (19). and θ0 [I(θ We now discuss the eﬀects of an increase in R&D productivity η. Fig. 5 (page 107) illustrates what happens in this case, both before and after unskilled workers (i.e., those who still remain unskilled after a vote on B for the new value of η) react by adjusting their desired level of ﬁring costs. The economy starts in the steady-state equilibrium E0 at which the resulting aggregate innovation rate happens to equal the innovation rate ˆ i < θ0 ). A positive desired by unskilled workers (the majority), I = I(θ R&D-technology shock +∆η (which was unexpected in the previous vote on B) shifts both the skilled labor full employment curve (19) and the R&D equilibrium curve (18) to the right, with the ordinate (abscissa) intercept of the latter (former) declining (rising). Without voting on ﬁring costs thereafter, the steady-state equilibrium would shift from E0 to E1 where the innovation rate has risen unambiguously, whereas the net eﬀect on the threshold ability level is ambiguous. The curve for (19) shifts outward since with a higher R&D productivity, any positive given amount (1 − θ02 )φN/2 of skilled workers can produce more R&D output than before (formally, θ0 rises for any given I > 0 in (19)). The downward shift of (18) is explained by the fact that expected R&D beneﬁts increase for given skilled labor input (these additional 28 By the second term in curly brackets of (23), consumption quantity is reduced due to higher markup pricing. This, however, does not aﬀect the optimal decision about B.

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q0 (18)

1

+DBˆ (q i < q 0 )

E2

+Dh

E0

E1

(r - n)×s × k hold × (l -1)× e-r× y

+Dh

(19)

Iˆ (q i < q 0 )

h old × f 2×k

h new × f 2×k

I

Fig. 5. Steady-state eﬀects of rising R&D productivity with voting response

beneﬁts will be transferred to a higher skilled wage rate because of free entry in the R&D race).29 However, since the welfare-maximizing innovation rate given in (24) is unaﬀected, unskilled workers (i.e., those individuals who decide not to invest in education under a vote on B for the given new η) raise their demand for employment protection in order to restore this level.30 Hence with voting on B, the curve for (18) rotates counterclockwise around its ordinate intercept like in Fig. 1, and the steady-state equilibrium moves from E1 to E2 . Relative to the old steady state at E0 , the proportion of unskilled workers has risen because fewer skilled workers than before are needed to produce ˆ i < θ0 ). Hence, in our diﬀerent the same aggregate innovation rate I(θ framework, Saint-Paul’s (2002, p. 699) conclusion that “[...] periods of high growth may be a more appropriate time for increasing labor market ﬂexibility [...]” 29 The formal proof is similar to Appendix A. Applying the implicit function theorem to (18) shows that dI/dη > 0 if and only if condition (11) is fulﬁlled, which holds necessarily for any positive R&D investment. Therefore, the R&D equilibrium curve shifts downward at any given level of θ0 . 30 ˆ i < θ0 )/∂η > 0 holds not only for our benchmark parameFormally, ∂ B(θ ter values but for the whole range of all possible parameter values that result ˆ i < θ0 ) > 0. in an interior steady-state solution with θ0 ∈]0, 1[ and B(θ

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no longer holds. Instead, we state our third main result: Proposition 5. Given I < I crit (I > I crit ), an exogenous positive growth shock – taking the form of a larger size of innovations λ or a higher R &D productivity η – increases political support of all workers (of the individuals who decide to stay unskilled under the new B) for larger ﬁring costs, and raises the proportion of unskilled workers in the new steady-state. In the case of a larger λ, the new desired innoˆ i < θ0 ) is lower than before, whereas this rate does not vation rate I(θ change in the case of a larger η. The results of this section could also be viewed as a caveat to the ﬁndings of Arnold (2002). He shows that in a North-South model of intraindustry trade with innovation and endogenous growth in the North and (exogenous) imitation by the South, the growth eﬀects of increased Southern imitation depend on Northern labor market ﬂexibility. The latter is modeled by an exogenous parameter denoting the outﬂow rate from unemployment which equals our job-ﬁnding rate p(ν). In the case of high (low) labor market ﬂexibility (i.e., large (low) p(ν)), increased Southern imitation stimulates (impedes) Northern economic growth. That is, alleviating the (only) labor market distortion in his model tends to support growth-enhancing eﬀects of a change to another exogenous parameter, in that case a rise in the Southern imitation rate.31 In our model, by contrast, alleviating the exogenous component of labor market rigidity (increase in A) creates political support for another, growth-retarding distortion of the labor market (an increase in ﬁring costs).32 This eﬀect oﬀsets any potentially growth-enhancing exogenous shocks (as discussed above) or policies like those related to international trade and imitation that could be added to our model: the steady-state utility growth rate with voting on ﬁring costs is ﬁxed ˆ i < θ0 ) log λ = ρ − n − β log λ. Therefore, there is no sysat z/z ˙ = I(θ tematic link anymore in our framework between one particular element 31 In Arnold’s model, the negative eﬀects of rising Southern imitation on Northern growth consist of a shortening of product life cycles (harming monopolists’ proﬁts and thus innovation incentives) and a rise in frictional unemployment (because imitation reallocates production to the South due to lower marginal labor costs). The positive eﬀect on Northern growth is rooted in the eﬃciency gain in Southern production, which raises world demand for Northern monopolists’ goods and thus their proﬁts, which stimulates innovation incentives. 32 A notable similarity between Arnold (2002) and our model is that more eﬃcient matching ceteris paribus (that is, for a given level of ﬁring costs in our case) raises the long-run growth rate. However, this result hinges crucially on the presence of scale eﬀects in Arnold’s model (a higher total number of employed workers implies an increase in the innovation rate), whereas our model is free of scale eﬀects (I is independent of the size of the total workforce).

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of exogenous labor market ﬂexibility and the innovation and growth rate. 6 Qualiﬁcations and Extensions 6.1 Accounting for Firm Ownership of Workers Neglecting ﬁrm ownership of workers inﬂuences obviously our results concerning the level of requested ﬁring costs. Like in Saint-Paul (2002), this assumption is made mainly because it allows us to concentrate on the conﬂicting interests among workers, instead of adding a political conﬂict between workers and shareholders (which may be identical to workers). Moreover, it allows us to solve our model explicitly for ˆ i < θ0 ) and to provide a clear discussion of its determinants. As in B(θ Saint-Paul’s model, taking account of ﬁrm ownership of workers would weaken political support for ﬁring costs (the optimal balance between quality growth and consumption quantity is found at a higher desired innovation rate), depending on the relative importance of labor and capital income. This is because the value of ﬁrms (thus, e−yr ϑ) depreciates with higher ﬁring costs. If the fraction of ﬁnancial assets in total income were higher for skilled workers than for the unskilled, this would decrease the demand for ﬁring costs among the former by more than among the latter. However, accounting for equal share ownership among skilled and unskilled workers aﬀects our results in (24) and (29) only quantitatively, without having any impact on Proposition 3. Moreover, Propositions 4 and 5 are not aﬀected at all, even when taking account of an unequal distribution of share ownership across worker types. Finally, since share ownership among workers is much more common in the US than in West European countries, this may provide one explanation for the much lower level of ﬁring costs observed in the US.

6.2 R &D Incentives of Industry Leaders One may wonder whether introducing ﬁring costs should provide an incentive for quality leader ﬁrms to improve their own products. By gaining a quality advantage of size λ2 instead of λ, they could try to avoid the risk of having to pay the ﬁring costs once being overtaken by an innovating follower ﬁrm. However, it turns out that this idea conﬂicts with the R&D process put forward in Sect. 2.2 above, which is taken from Segerstrom (1998) and Dinopoulos and Segerstrom (1999): it is assumed that all ﬁrms have the same R&D technology, and once a new innovation occurs, the patent attached to the previous leading technology expires and becomes common knowledge. Therefore, quality

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followers do not have to “catch up before they can pass” (i.e., they do not have to copy the current state-of-the-art product before being able to improve it), contrary to the R&D setup in Aghion et al. (2001).33 This means that even if quality leader ﬁrms improved their own products, the instantaneous probability of a further quality-augmenting innovation by followers would not change, hence the leader’s probability of having to pay the ﬁring costs does not change. The intuition follows Grossman and Helpman (1991): “[...] we implicitly suppose that potential entrants can, via inspection of the goods on the market, learn enough about the state of knowledge to mount their own research eﬀorts, even if the patent laws (or the lack of complete knowledge about best production methods) prevent them from manufacturing the current generation products. This speciﬁcation captures in part the often noted, public-good characteristics of technology”(ibid, p. 47). Hence, the “Arrow eﬀect” still ensures that industry leaders ﬁnd it more proﬁtable to direct their R&D investments toward other industries in which they are quality followers (see Grossman and Helpman, 1991, p. 47, for proof), irrespective of the ﬁring costs. To capture the empirical fact of signiﬁcant R&D expenditure of industry leaders, it would be necessary either to introduce R&D cost advantages for them relative to follower ﬁrms, as in Segerstrom and Zolnierek (1999) and Segerstrom (2004), or to assume Stackelberg behavior of industry leaders, as in Etro (2004). Both approaches, however, would complicate the Schumpeterian unemployment mechanism described in Sect. 2.3 considerably, which is beyond the scope of this paper (but might provide a fruitful subject for future research). 6.3 The Eﬀects of Firing Costs on Hiring A basic principle of labor economics is that higher ﬁring costs tend to decrease both hiring and ﬁring of ﬁrms, such that the net eﬀect on unemployment becomes ambiguous. This accords with empirical evidence referred to in footnote 1 above. At ﬁrst sight, it seems that this paper does not indicate the negative eﬀect of ﬁring costs on hiring: in (17), the steady-state vacancy rate is independent of B, and a higher level of B reduces unambiguously unskilled unemployment due to lower labor market turnover (declining “Schumpeterian unemployment”). Inspection of equation (13) that describes the process of vacancy creation 33

As noted by Segerstrom (2004), this assumption made by Aghion et al. (2001) has the counterfactual implication that small ﬁrms do not innovate.

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reveals that actually, we did account for the negative eﬀect of ﬁring costs on hiring: larger ﬁring costs reduce the aggregate innovation rate (since ﬁrms anticipate expected costs of layoﬀs when deciding on the optimal level of R&D investment), which tends to reduce the number of vacancies posted. However, this eﬀect is completely oﬀset by the induced increase in market size: since the unskilled unemployment rate declines and the proportion of unskilled workers rises with larger ﬁring costs, aggregate production must also increase, which raises unskilled labor demand of innovating ﬁrms. That is, larger ﬁring costs reduce the frequency of vacancy postings but once innovation occurs, the number of posted vacancies is higher, reﬂecting the larger market size for each incumbent monopolist. This is a knife-edge result that hinges crucially on the assumption of a linear production function Qt = (1 − u)θ0 Nt . Assume now that households use part of their savings to ﬁnance the accumulation of a physical capital stock K with Kt = K0 ent . Note that the capital stock must grow at the population growth rate in order to ensure constant per-capita consumption expenditure, similar to (6), that we need because of constant wage rates. With a Cobb-Douglas production function Qt = [(1 − u)θ0 Nt ]α Kt1−α ,

0 < α < 1,

the market-size eﬀect of a declining u and a rising θ0 would be alleviated, and a positive (negative) relationship between the steady-state vacancy rate and the aggregate innovation rate (the level of ﬁring costs) would result. To show this, we ﬁrst use the new production function (that equals total consumption) in (14) and solve this equation for the new steady-state unskilled unemployment rate in implicit form: u=

Ap(ν ∗ ) (1 − u)α K01−α I . +1− 1−α β βθ0

(32)

In (32), θ0 is determined by a new R&D equilibrium condition replacing (18), and by (19) for skilled labor full employment. Equation (32) reduces to (16) for α = 1, and it still implies a positive long-run relationship between innovation and unskilled unemployment. Then, using (32) and the new production function in (13) gives the new steady-state vacancy rate ∗

ν =

(1−u)α eny (I+β) θ01−α

β−δ

− Ap(ν ∗ )

K01−α ,

(33)

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which reduces again to (17) for α = 1. Now, an increase in ﬁring costs has two eﬀects on ν ∗ . First, by reducing the innovation rate, ν ∗ declines which is the expected negative eﬀect on hiring that vanishes in the linear-production case. Second, by reducing the unskilled unemployment rate, ν ∗ somewhat rises again. This second eﬀect is weakened indirectly, however, by the ﬁrst: a lower vacancy rate reduces the jobﬁnding rate p(ν ∗ ) due to the monotonous matching function and therefore tends to raise u according to (32) again. Overall, a net negative eﬀect of rising ﬁring costs on the steady-state vacancy rate will prevail, which somewhat weakens the market-size eﬀect referred to above. Hence, after a rise in ﬁring costs, the reduction (increase) in u (θ0 ) is alleviated, with the extent depending on the size of α. It turns out that this extension of the model does not aﬀect our main results qualitatively. The crucial point in the logic of Proposition 3 is that for I < I crit , skilled workers realize additional beneﬁts from higher ﬁring costs relative to the unskilled because the education equilibrium eﬀect dominates the skilled labor demand eﬀect. This reasoning and thus Proposition 3 still hold. Proposition 4 also remains valid since dI/dA > 0 and dI/dB < 0 continue to hold. All workers still face the same trade-oﬀ between consumption quantity and consumption quality growth. However, with the new production function, the increase in consumption quantity after a marginal rise in ﬁring costs is smaller than before. This is because the negative eﬀects of ﬁring costs on hiring alleviate the net reduction in unskilled unemployment. Since the negative eﬀect of a marginal reduction in the innovation rate on workers’ utility in terms of quality growth is unchanged, they vote for a lower level of B than in the linear-production case. Hence, for a given output ˆ i < θ0 ) that is realized after the vote on level, the innovation rate I(θ B is larger than with the old production function. Finally, by a similar argument, the logic of Proposition 5 is also not aﬀected qualitatively by this extension of our model.

6.4 Accounting for Repeated Voting In general, with repeated voting on a policy issue, there may be important general-equilibrium feedback eﬀects that aﬀect voting behavior in the future, which, with forward-looking voters, can inﬂuence voting behavior today. In particular, rational voters will take into consideration that their vote today may change the distribution of workers tomorrow, which could induce strategic voting. This in turn could lead to situations of multiple politico-economic equilibria. In their introductory section, Hassler et al. (2003) provide a broad overview over the relevant literature on these issues.

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Fortunately, things are much simpler in our framework. First of all, since all unskilled and all skilled workers have the same preferences about ﬁring costs, respectively, there are only two possible political ˆ i < θ0 ), preferred by all unskilled workers and given in outcomes: B(θ ˆ i < θ0 ), preferred by all unskilled workers if ˆ (25), and B(θi ≥ θ0 ) > B(θ crit ˆ i ≥ θ0 ) = 0 if I ≥ I crit ). Second, these (or, alternatively, B(θ I 0.5 holds (which can be stated in terms of fundamental parameters, see footnote 21). 7 Conclusions This paper contributes to the literature on the growth eﬀects of labor market ﬂexibility from a political economy point of view within the setup of a standard Schumpeterian endogenous growth model without scale eﬀects. Given the rigidity of non-instantaneous matching of unskilled workers with vacancies posted by new quality leader ﬁrms, we analyze both general equilibrium consequences of and political demand for ﬁring costs. We ﬁnd that although ﬁring costs reduce the steadystate innovation and growth rate, all workers vote for positive ﬁring costs (if the innovation rate falls short of a critical level) by evaluating a trade-oﬀ between the quantity of consumption and the growth of the quality of consumer goods. If the innovation rate is below the critical level, skilled workers (i.e., those individuals who still invest in education after the majority-winning unskilled workers have decided on ﬁring costs) even vote for larger ﬁring costs than unskilled workers, although only the latter can become unemployed. Moreover, we ﬁnd that workers increase their demand for ﬁring costs in the case of more eﬃcient matching on the labor market. Hence, alleviating a labor market rigidity that is exogenous to our framework aggravates the rigidity that is determined endogenously by majority voting. We can derive two major insights from our analysis. First, contrary to the conclusion of Saint-Paul (2002), our results do not support the view that employment protection is likely to be lower in fast growing economies. Once the aggregate innovation and growth rate is made endogenous in a neo-Schumpeterian framework, all workers have an individually ‘optimal’ rate of creative destruction. The median voter’s (single-peaked) preferences for this rate of innovation are realized by adjusting the endogenous labor market rigidity accordingly. If the exogenous component of labor market rigidity or other exogenous

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parameters change in a way to increase innovation and long-run growth, workers vote for more employment protection. Second, our results suggest that the interplay between labor market ﬂexibility and economic growth tends to be more complicated than reﬂected in popular statements like ‘a ﬂexible labor market tends to support economic growth’. This may well hold for all deﬁning elements of labor market ﬂexibility taken as an aggregate (hence it holds in the model of Arnold, 2002, where there is just one type of labor market rigidity). However, because some of those elements are endogenous to the political process and interact with other elements as illustrated for example by our model, it seems to be questionable to explain cross-country variations in economic growth by diﬀerences in a single aspect of labor market rigidity. Our results are unlikely to change qualitatively once one extends the analysis to cover the case of incomplete unemployment insurance. This would introduce unemployed workers’ preferences for ﬁring costs that diﬀer from those of unskilled employed workers. However, it seems justiﬁable to assume that the median voter is still an unskilled employed worker who mainly cares about expected wage earnings.34 An interesting extension of our analysis would be to add the possibility for ﬁrms to escape ﬁring costs by outsourcing the production of consumer goods to a foreign country. Alternatively, instead of ﬁring costs, one could also analyze the preferences of diﬀerent worker groups for unemployment beneﬁts within our dynamic setting. Appendix A. The Proof of dI/dθ0 > 0 in (18) To prove that the curve for (18) is upward sloping in Fig. 1, we write

f (I, θ0 ) ≡ I −

ηe−ρy Ap(ν ∗ )θ02 (λ − 1) − (ρ − n)σAp(ν ∗ )k = 0 (A. 1) σAp(ν ∗ )k + (β + I)ηe−ρy θ0 B

34 Saint-Paul (1996) distinguishes ﬁve groups of individuals (skilled workers, unskilled and semi-skilled workers, short-term unemployed, long-term unemployed, and capitalists) and argues that

“typically, the decisive voter will be considered as a member of [the] group [of] unskilled and semi-skilled employed workers. This is meant to be a relatively broad group, including workers without a college degree, who make up more than 70% of the workforce in most European countries” (ibid, p. 275).

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and apply the implicit function theorem: ∂f /∂θ0 dI =− ∂f /∂I dθ0 −[2ηe−ρy Ap(ν ∗ )θ0 (λ − 1)]den + nom(β + I)e−ρy ηB , =− den2 + nom e−ρy θ0 ηB (A. 2) where nom and den denote nominator and denominator of (18), respectively. Using den/nom = 1/I, (A.2) takes a positive value if and only if 2Ap(ν ∗ )θ0 (λ − 1) > I(β + I)B (A. 3) is fulﬁlled. By using β +I = Ap(ν ∗ )/(1−u) from (16) and subsequently θ0 (1 − u) = c/λ from (6), condition (A.3) can be restated as follows: 2c(λ − 1)/λ > IB.

(A. 4)

This is fulﬁlled because of our assumption (11). B. Proof of Proposition 1 Equation (18) can be solved for

Ap(ν ∗ ) (ρ − n)σk σAp(ν ∗ )k = 0. (A. 5) − θ (λ − 1) +β I+ I 2 + −ρy 0 e−ρy ηθ0 B e ηθ0 B . ,+ . ,+ ≡Q

≡P

This quadratic equation yields two roots 0 I1,2 = −P/2 ± (P/2)2 − Q, of which only I1 = −P/2 +

0

(P/2)2 − Q

is positive. Moreover, there are no complex solutions since Q < 0 immediately follows from looking at the ordinate intercept of (18) in Fig. 1. Thus, I1 is the only feasible solution. Diﬀerentiating this with respect to B gives 1 Q β−P 1 P −β dI1 . (A. 6) + + [(P/2)2 − Q]− 2 P = B 2B 2 2B dB The derivative in (A.6) is negative if and only if 1

[(P/2)2 − Q]− 2

P (P − β) − Q > P − β, 2

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which can be rewritten as 1 P Q P > [(P/2)2 − Q] 2 = I + , − 2 P −β 2

(A. 7)

where the equality on the RHS follows from our deﬁnition in (A.5). Simplifying and substituting for P and Q in (A.7) ﬁnally gives the condition e−ρy ηθ02 (λ − 1) − (ρ − n). (A. 8) I< σk Note that the RHS of (A.8) equals (18) for B = 0. Since (18) can be written as e−ρy ηθ02 (λ−1) − (ρ − n) σk , (A. 9) I= (β+I)e−ρy ηθ0 B 1 + σAp(ν ∗ )k (A.8) follows immediately. Hence, we have shown that an increase in ﬁring costs reduces the value for the innovation rate I deﬁned implicitly in (18), given any feasible value for θ0 . Thus, the curve for (18) shifts to the left as depicted in Fig. 1, whereas the curve for (19) is not aﬀected. This proves Proposition 1. C. The Proof of dI/dy < 0 in (18) In order to show that the curve for (18) shifts downward again after the induced decline in the matching time y as shown in Fig. 3, we use (A.1) and implicit diﬀerentiation: ∂f /∂y dI = ∂f /∂I dy −[−ρηe−ρy Ap(ν ∗ )θ02 (λ − 1)]den + nom[−ρ(β + I)ηe−ρy θ0 B] . =− den2 + nom e−ρy θ0 ηB (A. 10) Using den/nom = 1/I, (A.10) takes a negative value if and only if c(λ − 1)/λ > IB is satisﬁed, which is our assumption in (11). Therefore, a reduction in y shifts the curve for (18) downward in Fig. 3, whereas the curve for (19) is not aﬀected. Thus, dI/dy < 0 follows immediately.

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Acknowledgements The author would like to thank seminar participants of the University of Dortmund (in particular Christian Bayer), participants of the DEGIT VIII conference, Helsinki, May 30-31, 2003 (in particular Volker Grossmann, Ingrid Kubin and Omar Licandro), and participants of the annual meeting of Verein f¨ ur Socialpolitik, Zurich, September 30 - October 3, 2003, for helpful discussion. The constructive comments of an anonymous referee are gratefully acknowledged. The usual disclaimer applies.

References Aghion, P., Harris, C., Howitt, P., and Vickers, J. (2001): “Competition, Imitation and Growth with Step-by-Step Innovation.” Review of Economic Studies 68: 467-492. Aghion, P., and Howitt, P. (1992): “A Model of Growth through Creative Destruction.” Econometrica 60: 323-351. Aghion,P., and Howitt, P. (1994): “Growth and Unemployment.” Review of Economic Studies 61: 477-494. Aghion, P., and Howitt, P. (1998): Endogenous Growth Theory. Cambridge, Mass.: MIT Press. Arnold, L. G. (2002): “On the Growth Eﬀects of North-South Trade: the Role of Labor Market Flexibility.” Journal of International Economics 58: 451-466. Dinopoulos, E., and Segerstrom, P. S. (1999): “A Schumpeterian Model of Protection and Relative Wages.” American Economic Review 89: 450-472. Dinopoulos, E., and Thompson, P. (1996): “A Contribution to the Empirics of Endogenous Growth.” Eastern Economic Journal 22: 389-400. Etro, F. (2004): “Innovation by Leaders.” Economic Journal 114: 281- 303. Grossman, G. M., and Helpman, E. (1991): “Quality Ladders in the Theory of Growth.” Review of Economic Studies 58: 43-61. Hassler, J., Rodriguez Mora, J. V., Storesletten, K., and Zilibotti, F. (2003): “The Survival of the Welfare State.” American Economic Review 93: 87-112. Jones, C. I. (1995): “R&D-Based Models of Economic Growth.” Journal of Political Economy 103: 759-784. Jones, C. I. (1999): “Growth: With or Without Scale Eﬀects?” American Economic Review, Papers and Proceedings 89: 139-144. Joseph, G., and Weitzenblum, T. (2003): “Optimal Unemployment Insurance: Transitional Dynamics vs. Steady State.” Review of Economic Dynamics 6: 869-884. OECD (1999): Employment Outlook. Paris. OECD (2003): Education at a Glance. Paris.

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Reinberg, A., and Hummel, M. (2003): “Geringqualiﬁzierte: In der Krise verdr¨ angt, sogar im Boom vergessen.” IAB Kurzbericht Nr. 19, Institut f¨ ur Arbeits- und Berufsforschung der Bundesanstalt f¨ ur Arbeit, Germany. Saint-Paul, G. (1996): “Exploring the Political Economy of Labor Market Institutions.” Economic Policy: A European Forum 23: 263-315. Saint-Paul, G. (2002): “The Political Economy of Employment Protection.” Journal of Political Economy 110: 672-704. Segerstrom, P. S. (1998): “Endogenous Growth Without Scale Eﬀects.” American Economic Review 88: 1290-1310. Segerstrom, P. S. (2004): Intel Economics. Mimeo, Stockholm School of Economics. Segerstrom, P. S., Anant, T. C. A., and Dinopoulos, E. (1990): “A Schumpeterian Model of the Product Life Cycle.” American Economic Review 80: 1077-1092. Segerstrom, P. S., and Zolnierek, J. (1999): “The R&D Incentives of Industry Leaders.” International Economic Review 40: 745-766. S ¸ ener, M. F. (2000): “A Schumpeterian Model of Equilibrium Unemployment and Labor Turnover.” Journal of Evolutionary Economics 10: 557-583. S ¸ ener, M. F. (2001): “Schumpeterian Unemployment, Trade and Wages.” Journal of International Economics 54: 119-148. Siebert, H. (1997): “Labor Market Rigidities: At the Root of Unemployment in Europe.” Journal of Economic Perspectives 11: 37-54. Steger, Thomas M. (2003): “The Segerstrom Model: Stability, Speed of Convergence and Policy Implications.” Economics Bulletin 15: 1-8. Wolﬁnger, R. E., and Rosenstone, S. (1980): Who Votes? New Haven: Yale University Press. Address of author: – Wolf-Heimo Grieben, University of Dortmund, Department of Economics, Vogelpothsweg 87, 44221 Dortmund, Germany (email: [email protected])

J. Econ. (2005) Suppl. 10: 119-142

White-Collar Employment, Inequality, and Technological Change Volker Grossmann Received January 29, 2003; Revised version received July 19, 2004 c Springer-Verlag 2005

This paper develops a R&D-based growth model to examine the relationship between technological change, growth, welfare and the demand for skillintensive, analytical activities (e.g., product development, quality-control, and design of advertising campaigns). Results are consistent with evidence on rising employment shares of skilled, white-collar workers and increases in the skill premium in the US or UK. Moreover, accounting for a simultaneous decrease in overhead labor requirements (e.g., administrative staﬀ), the analysis suggests that recent technology shifts have no systematic impact on ﬁrm sizes and on the economy’s rate of growth. This sheds some light into the “Solow-productivity paradox”. Keywords: analytical skills, skill premium, R&D-based growth, white-collar employment JEL classiﬁcation: O31, O33, J21, J31.

1 Introduction There is now a broad consensus that in the 1980s and 1990s the structure of labor demand has shifted in favor of skilled workers, and that this shift can largely be attributed to technological advances. However, understanding the interplay between new technologies and the structure of labor demand requires to shed light into the decision of ﬁrms how to allocate workers with diﬀerent skills to diﬀerent tasks. For instance, contrary to earlier notions of “skill-biased technological change” (SBTC), computer users per se do not seem to have gained more from computerization than workers using a pencil (DiNardo and Pischke, 1997). In fact, administrative workers, who nowadays use computers more intensively than most other group, seem to be clear losers of the recent technological revolution (e.g. Berman, Bound and Grichilis,

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1994). Such evidence demonstrates that the mere observation that shifts in the labor demand structure are related to the emergence of new information and communication technologies (ITC) is not a very useful hypothesis. This paper attempts to draw a more diﬀerentiated picture of the relationship between technological change, growth and the structure of employment by hypothesizing that computerization has favored skillintensive, analytical activities like product development, quality-control, design of customer services, and promotion of products by advertising campaigns. These tasks require much analytical thinking and are based on eﬃcient ﬂows of information about markets and customers. Creation and analysis of such information are favored by new ITC. New ITC allow marketing managers to assemble, store and analyze customer data like demographics and purchase habits (‘data mining’). In turn, ‘data warehouses’ enable ﬁrms to design and keep track of marketing campaigns and to target consumers more eﬀectively than by mass-media advertising (e.g. Bresnahan, 1999; Shapiro and Varian, 1999). Moreover, enhanced possibilities to do research on consumers’ preferences, the emergence of computer-aided design and more eﬃcient interactions between design, production and marketing help ﬁrms to improve the quality of products. I develop a non-scale endogenous growth model which allows for both types of demand-enhancing tasks, R&D, performed in-house in order to improve the quality of goods, and advertising, which is viewed as promotional activity.1 Firms can freely enter the economy but have to cover the costs for non-production labor from subsequent proﬁts under monopolistic competition. This is because R&D and advertising activities are incurred prior to product market competition, and thus give rise to endogenous sunk cost. Moreover, ﬁrms have to incur ﬁxed overhead labor requirements in terms of both skilled or unskilled workers, which may be interpreted to include administrative staﬀ. First, I show that an increase in the eﬀectiveness of skill-intensive, quality-improving (R&D) or promotional activities fosters a reallocation of skilled workers from production-related activities towards these analytical, demand-enhancing tasks. This shift in the employment structure is consistent with evidence provided by the empirical literature on SBTC, which shows a clear upward trend in the share of skilled, white-collar workers like managers and professionals (e.g., Berman, Bound and Grichilis, 199